Uncovering the High Scale Higgs Singlet Model
UUncovering the High Scale Higgs Singlet Model
Sally Dawson, Pier Paolo Giardino, and Samuel Homiller Department of Physics, Brookhaven National Laboratory, Upton, N.Y., 11973, U.S.A. Instituto Galego de F´ısica de Altas Enerx´ıas, Universidade de Santiago de Compostela,15782 Santiago de Compostela, Galicia, Spain Department of Physics, Harvard University, Cambridge, MA, 02138, U.S.A. (Dated: February 8, 2021)
Abstract
The scalar singlet model extends the Standard Model with the addition of a new gauge singletscalar. We re-examine the limits on the new scalar from oblique parameter fits and from a global fitto precision electroweak observables and present analytic expressions for our results. For the casewhen the new scalar is much heavier than the weak scale, we map the model onto the dimension-six Standard Model effective field theory (SMEFT) and review the allowed parameter space fromunitarity considerations and from the requirement that the electroweak minimum be stable. Aglobal fit to precision electroweak data, along with LHC observables, is used to constrain theparameters of the high scale singlet model and we determine the numerical effects of performingthe matching at both tree level and 1-loop. a r X i v : . [ h e p - ph ] F e b . INTRODUCTION The Higgs singlet model [1–7] has been extensively studied as a simple extension ofthe Standard Model (SM) containing only one new particle. Depending on the potentialparameters, the model can lead to a first order electroweak phase transition [8–18], makingit highly motivated in addressing the problem of baryogenesis. It can also arise as thelimiting case of many interesting models addressing the hierarchy problem [19, 20] or evendark matter [21–27]. When the mass of the new scalar becomes much larger than the weakscale, the theory can be mapped onto an effective field theory. The utility and simplicityof the model thus makes it an ideal candidate for exploring the limits of an effective fieldtheory framework in reproducing the features of the underlying UV models [28–35].In the full UV complete singlet model, restrictions on the parameters can be found fromfits to precision electroweak observables as well as LHC data. These limits can then becompared with limits found in the context of a low energy effective field theory. We consideran effective field theory in which the SM Higgs doublet is constrained to be an SU (2) doublet,the Standard Model effective field theory (SMEFT). At tree level, the singlet model generatesonly two SMEFT coefficients when matched at the UV scale [31, 36]. The aim of this workis to examine to what extent the extraction of SMEFT coefficients from global fits at theweak scale gives information on the parameters of the UV complete singlet model [28, 32–34, 37–39]. The focus is on understanding the numerical importance of various choices madewhen performing the low energy fits and to this end, we implement both tree and 1-loopmatching [40–42] at the UV scale. We find that the effects of the 1-loop matching aretypically rather small. Effects of O (10%) can be obtained only for rather large values ofcertain dimensionless parameters in the Lagrangian.Section II contains a recap of the model and restrictions on the model parameters fromunitarity and the minimization of the potential. Analytic results for electroweak precisionobservables in the singlet model are found in Section III along with a comparison between aglobal fit to electroweak precision observables (EWPOs) and a fit to the oblique parameters,and restrictions from unitarity and the minimization of the potential are in Section IV. TheSMEFT matching with the singlet model at both tree and loop level is studied in Section Vand a global fit to electroweak precision observables, Higgs, and di-boson data is presented.Section VI has some conclusions. 2 I. BASICS
The singlet model we consider contains the SM Higgs doublet, Φ, and a scalar gaugesinglet, S . The most general scalar potential is, V (Φ , S ) = − µ H Φ † Φ + λ H (Φ † Φ) + m ξ † Φ S + κ † Φ S + t S S + M S + m ζ S + λ S S . (1)The parameters can be redefined such that (cid:104) S (cid:105) ≡ x = 0. After spontaneous symmetrybreaking, the 2 neutral scalars, Φ and S , mix to form the physical scalars, h and H , h = cos θ Φ + sin θ SH = − sin θ Φ + cos θ S , (2)with the physical masses, m h = 125 . M H . The parameters of the model can betaken as, m h , M H , v = 246 GeV , sin θ, x = 0 , κ, m ζ , λ S . (3)The other parameters of the Lagrangian are determined in the singlet model by: m ξ = m h − M H v sin 2 θ ,M = m h sin θ + M H cos θ − κ v ,λ H = m h cos θ + M H sin θ v . (4)The Z symmetric case has m ζ = t S = m ξ = 0 and x (cid:54) = 0.The couplings of h to SM fermions and gauge bosons are suppressed relative to the SMHiggs couplings by a factor of cos θ , while the H couplings are suppressed by sin θ . Wecan thus immediately find a trivial limit on cos θ from Higgs production to SM particles X ,(assuming no decays to invisible particles), cos θ = µ ≡ σ · BR( σ · BR) SM (5) We note that for κv (cid:29) M , the mass of the new scalar, M H , comes from electroweak symmetry breakingand in this case the theory cannot be mapped onto the SMEFT [42, 43]. Additionally, the kinematicdistributions for hh production in this limit are quite different from those where M H primarily dependson M [44]. If 2 M H < m h then the decay h → HH is allowed, altering the limit on cos θ . − [45] and the CMS combinedlimits with 139 fb − [46], µ [ATLAS] = 1 . + . − . , µ [CMS] = 1 . + . − . (6)we find at 95% C.L., | sin θ | < . m h < M H . (7)For m h > M H , the naive limit of Eq. (7) does not apply because the h decays to HH must be included and this branching ratio is sensitive to the other parameters of the scalarpotential. Limits on the singlet model from resonant double Higgs production are beginningto be competitive with those from single Higgs production for M H (cid:46)
700 GeV [47], althoughour primary focus here will be on M H ∼ (1 −
2) TeV.
III. RESTRICTIONS ON MODEL PARAMETERS
The parameters of the singlet model can be limited by a fit to the Z - and W -pole ob-servables (we term this the EWPO fit): M W , Γ W , Γ Z , σ h , R b , R c , R l , A F B,b , A
F B,c , A
F B,l , A b , A c , A l . (8)The SM results for these observables are well known [48, 49]. In a previous study, Ref. [50],we computed the limits on the coefficients of an effective field theory that result from a fitto the observables of Eq. (8) computed to NLO in both QCD and electroweak interactions,and we apply an identical calculational framework here. The observables and SM theorynumbers used in the current study can be found in in Table III of Ref. [50]. We takeas our input parameters: G µ = 1 . × − GeV − , M Z = 91 . ± . /α = 137 . α (5)had = 0 . ± . α s ( M Z ) = 0 . ± . m h =125 . ± .
14 GeV, m b = 4 .
18 GeV and M t = 172 . ± . G µ and the vacuum expectation value v is, as usual, G µ = 1 √ v (1 + ∆ r ) (9)where, ∆ r = ∆ r SM + ∆ r singlet .
4n computing ∆ r , we use ˆ M W ≡ ( M Z / (cid:18) (cid:113) − √ παG µ M Z (cid:19) calculated from our inputs. Forsimplicity, we define h ≡ m h , H ≡ M H , z ≡ M Z and w ≡ ˆ M W and obtain the simple form, ∆ r singlet = ∆ r singlet ( h, H ) − ∆ r singlet ( H, h )∆ r singlet ( h, H ) = √ θG µ π (cid:26) − h wA ( h )( h − w ) + 3 whA ( w )( H − w )( h − w ) (cid:27) . (10)We find the one-loop prediction for M W in the singlet model, M W = M SM W + F W ( h, H ) − F W ( H, h ) F W ( h, H ) = α sin θ πM W (cid:26) hz w − z ) + A ( h )12( h − w ) w (2 w − z ) (cid:18) hw ( w − z ) + 12 w z + h ( z − w ) (cid:19) + hzA ( w )12( h − w ) w ( H − w )( z − w )(2 w − z ) (cid:18) (2 w − z ) (cid:20) hH − w ( h + H ) (cid:21) + w (8 z − w ) (cid:19) + hwA ( z )12( z − w )( z − w ) − zB ( w, h, w )12 w ( w − z ) (cid:18) h − hw + 12 w (cid:19) + wB ( z, h, z )12( z − w )( z − w ) (cid:18) h − hz + 12 z (cid:19)(cid:27) . (11)This is in agreement with Ref. [52]. For a massless b quark, the total W decay width is,Γ W = Γ SM W + G W ( h, H ) − G W ( H, h ) G W ( h, H ) = − √ wG F sin θ π (cid:26) wh ( − hH − w + 4 wh + 4 wH )2( h − w )( H − w ) − A ( h ) h ( h − w )( h − w ) (cid:18) h − h w + 21 h w − hw + 8 w (cid:19) + hA ( w )( h − w )( h − w )( H − w )( H − w ) (cid:18) w ( H + h ) − whH ( H + h )+ h H − w ( H + h ) + 18 w hH − w (cid:19) + B ( w, h, w )( h − w ) (cid:18) h − h w + 20 hw − w (cid:19)(cid:27) . (12) The function A is defined as A ( m ) = (cid:90) d d k (2 π ) k − m , where we calculate in d = 4 − (cid:15) dimensions. B is the Passarino- Veltman 2-point function, B ( m , m , p ) = (cid:90) d d k (2 π ) n k − m ][( k + p ) − m ] . The Passarino-Veltman functions are evaluated using
QCDLOOPS [51]. b mass contribution to Z decays to bottom pairs is sensitive to the Higgs- b Yukawa coupling and generates non-oblique contributions. We compute R l , R b , R c and Γ Z for m b (cid:54) = 0 and find that the numerical effect is less than ∼
2% for M H >
20 GeV, rising to ∼
5% for M H ∼
10 GeV, justifying the neglect of b mass effects in our fits.We perform a fit, including correlations, to the observables of Eq. (8) to determine themaximum allowed value of sin θ for a given value of M H including all one-loop contributions.It is of interest to compare the complete EWPO fit with the results using the obliqueparameters only. Using the results of [53, 54], we find that the differences between thePeskin-Takeuchi [55] variables in the Higgs singlet model and the SM take the form,∆ S = sin θ π ( G ( H, z ) − G ( h, z )) (13)∆ T = 3 sin θ πs W c W ( K ( H ) − K ( h )) (14)∆ S + ∆ U = sin θ πc W ( G ( H, w ) − G ( h, w )) , (15)where s W = 1 − c W = 1 − w/z is sin θ W of the electroweak mixing angle and we define, K ( h ) = h ( ( z − w )( h − w )( h − z ) A ( h ) + A ( w )( h − w ) − A ( z )( h − z ) ) (16) G ( h, z ) = h F ( h, z )( A ( h ) − A ( z ) − ( h − z ) B ( z, h, z )) (17) F ( h, z ) = h − hz + 12 z z ( h − z ) . (18)We fit to the values in [56], ∆ S = − . ± . T = 0 . ± . U = 0 . ± .
11 (19)with the correlation matrix, ρ = . . − . .
92 1 . − . − . − .
93 1 . . (20)In Fig. 1 we report the results corresponding to different sets of observables:6 M H (GeV) | s i n θ | m a x EWPO FitM W onlyZ pole observables onlySTUSinglet Model Maximum allowed sin θ
200 400 600 800 1000 M H (GeV) | s i n θ | m a x EWPO FitM W onlyZ pole observables onlySTUHiggs CouplingsSinglet Model Maximum allowed sin θ FIG. 1: Maximum allowed sin θ in the Higgs singlet model at 95% confidence level based on fits toelectroweak precision observables (EWPO), the W mass measurement only, the Z pole observablesonly, and to the ST U parameters as described in the text. A naive limit from the Higgs couplingmeasurements is shown on the RHS for comparison. • Only M W • The Z pole observables alone • Oblique parameters only • EWPOs given in Eq. (8).The results for the fit to M W alone are in agreement with those of Ref. [57] and are agood approximation to the complete EWPO fit. The EWPO fit limits are in agreementwith Ref. [39] after adjusting for the different input parameters. It is interesting that thecurrent limits from Higgs couplings give better bounds for all M H (cid:46) For the case where the second We find rough agreement with Refs. [12, 39] (the differences can be explained by the different numericalvalues of the input parameters) and disagree with the oblique parameter limits of Fig. 1 of [58]. We notethat the curve labelled “Exact Singlet” on the RHS of Fig. 1 of Ref. [28] is the
ST U result and has useda slightly different fit to the oblique parameters [59] from the PDG [56] results used here. The curvelabelled Higgs in that plot is the prediction from fitting Higgs data within the context a SMEFT fit andthus differs from the SM Higgs coupling fit shown in Fig. 1. M H < m h , the limits obtained from the oblique parameters are not a goodapproximation of the complete EWPO fit. IV. THEORETICAL CONSTRAINTS
In Section V, we will match the singlet model with a very heavy H to the SMEFT. Beforewe do so, we consider the theoretical restrictions on the singlet model parameters that arerelevant for the matching. A. Vacuum Structure of the Potential
The first set of theoretical constraints on the singlet model come from requiring a suitablevacuum structure of the potential [5, 7, 10, 60]. Demanding that the potential is stable atlarge field values leads to the requirement λ H , λ S >
0, and κ ≥ − √ λ H λ S [7], where λ H is determined by Eq. (4). Additional bounds result from requiring that the electroweakminimum be the global minimum of the potential. Following [7], we compute these boundsby finding all the extrema of the potential expanded around the electroweak vev as a functionof ( v, x ), and then checking whether or not the value of the potential at ( v = 246 GeV , x = 0)is the global minimum.The extrema of the potential can be divided into two classes: extrema where v (cid:54) = 0, andthose where v = 0. In the former case, the new extrema are denoted ( v ± , x ± ) as in ref. [7],and tend to bound lower values of κ . In the latter case, the extrema are denoted by (0 , x ± ),and these tend to limit both large values of κ as well as large values of m ζ . An exampleof the vacuum structure is shown in Fig. 2, where we illustrate the regions excluded by theemergence of different global minima as well as the condition from vacuum stability. Fig. 3illustrates how these bounds change as a function of the physical parameters. In particular,we see that for larger masses, M , the bounds on m ζ /M from the v = 0 minima becomeconstant as a function of κ , depending only on λ S . It is interesting that quite large valuesof κ are allowed in all scenarios. The upper bound on m ζ /M never exceeds O (2 − � � � � � � � � � � � � �� � � ��� ��������� ( � ± � � ± ) ��� ������������ � = � FIG. 2: Demonstration of the bounds from the appearance of other global minima in the κ vs. m ζ /M plane for M = 2 TeV, sin θ = 0 .
25, and λ S = 1 . B. Unitarity
The next set of theoretical constraints come from the requirements of tree-level pertur-bative unitarity [5, 61–63]. The simplest constraints come from hh → hh and HH → HH scattering, where the spin-0 partial waves in the high energy limit are a ( hh → hh ) (cid:12)(cid:12)(cid:12) s (cid:29) m h = − π (cid:0) λ H cos θ + κ sin θ cos θ + λ S sin θ (cid:1) (21) a ( HH → HH ) (cid:12)(cid:12)(cid:12) s (cid:29) m H = − π (cid:0) λ S cos θ + κ sin θ cos θ + λ H sin θ (cid:1) (22)For sin θ (cid:28)
1, requiring | a | < / λ S , λ H (cid:46) π/
3. This bound on λ H indirectly bounds M H as a function of sin θ : M H sin θ (cid:46) π v − m h cos θM H (cid:46) θ = 0 . . (23)The similar bound from hH → hH scattering only restricts | κ | (cid:46) π .9 IG. 3: Regions in the singlet model where the electroweak minimum is the global minimum ofthe potential as a function of κ and m ζ /M , varying the other physical parameters. . ONE-LOOP MATCHING OF THE SINGLET MODEL TO SMEFTA. One-loop Matching When the mass of the heavy scalar is much larger than the weak scale and any relevantenergy scales, the singlet model can be modeled by an effective field theory, L singlet −−−−−→ M H →∞ L SM + (cid:88) i C i ( M ) M O (6) i + . . . (24)with coefficients matched to the singlet model at the high scale, M . We retain only thedimension-6 operators, O (6) i ,and use the Warsaw basis [64] with the notation of Ref. [65].The global fits of Ref. [28] were performed using tree level matching at the scale M . It is of interest to implement the one-loop matching for the case of the singlet model andexamine the numerical impacts. The coefficients at the matching scale, M , generically takethe form, C i ( M ) = c i ( M ) + d i ( M )16 π , (25)where c i ( M ) is the tree level result and d i ( M ) / (16 π ) is the one-loop contribution at thematching scale. When the renormalization group evolution to the low scale µ R is included, C i ( µ R ) = c i ( M ) + d i ( M )16 π + γ ij π c ij ( M ) log (cid:18) µ R M (cid:19) . (26)In the case of the singlet model only two coefficients are generated at tree level [31, 36, 63, 66], c H (cid:3) = − m ξ M (27) c H = m ξ M (cid:18) m ξ m ζ M − κ (cid:19) , (28)with all other c i ( M ) = 0. However, there are many coefficients generated at one-loop atthe matching scale, M [40–42]. The majority of these coefficients are proportional to thetree level coefficient, c H (cid:3) . We use the shorthand C Hu → C Hu , C Hc , C Ht , etc., and take y u = y c = 0 , y t = M t √ /v (similarly we set all other y i = 0) and we further assume Ref. [29] noted that better agreement between the SMEFT and singlet model predictions for hh productionare obtained when the matching is performed at the physical mass, M H . The one-loop matching wouldthen contain terms proportional to log( M H /M ) that we have omitted. C (1) Hq , C (3) Hq , C (1) Hl , and C (3) Hl are flavor diagonal and use an analogous shorthand. Forconvenience, we list the results of Ref. [40] in our notation: d HD = 31 g (cid:48) c H (cid:3) d HW = − g c H (cid:3) d HB = − g (cid:48) c H (cid:3) d HW B = − gg (cid:48) c H (cid:3) d Hu = 1108 (34 g (cid:48) − y u ) c H (cid:3) d Hd = 13 d He = 23 d (1) Hl = − g (cid:48) c H (cid:3) d (1) Hq = 1216 (17 g (cid:48) + 135 y u ) c H (cid:3) d (3) Hq = 172 (cid:20) g − y u (cid:21) c H (cid:3) d (3) Hl = 17 g c H (cid:3) d y = − c H (cid:3) . (29)The one-loop contribution d tH can be written in terms of c H (cid:3) and C H and is, d tH = y t (cid:20) −
118 (45 y t − g ) c H (cid:3) + 32 c H − λ c H (cid:3) (cid:21) , (30)where in the SMEFT the physical Higgs mass is determined in terms of the potential pa-rameters to O ( v /M ) by [65], m h v = λ H (cid:18) v M c H (cid:3) (cid:19) − v M c H (31)and we define, λ ≡ λ H (cid:18) v M c H (cid:3) (cid:19) λ = m h v + 3 v M c H + O (cid:18) v M (cid:19) , (32)where we note that Ref. [40] absorbs the factor of c H (cid:3) into the definition of λ used inthe matching conditions, along with a relative factor of 2 in the definition of the quartic Since y t is the only non-zero Yukawa that we include, O y = y t tttt . Eq. (31) represents the dimension-6 SMEFT limit of Eq. (4) for therelationship between the parameters of the potential and m h .Finally, the coefficients generated at tree level also receive one-loop corrections, d H (cid:3) = − λc H (cid:3) + 3136 (3 g + g (cid:48) ) c H (cid:3) + 32 c H + δd H (cid:3) + δd shiftH (cid:3) d H = λ (cid:20)(cid:18) g − λ (cid:19) c H (cid:3) + 6 c H (cid:21) + δd H + δd shiftH . (33)where, δd H = − κ
12 + m ξ M (cid:18) m ξ (cid:0) λ − κλ + 112 κ − κλ S (cid:1) − κ m ζ (cid:19) + m ξ M (cid:18) m ξ (cid:0) κ − λ S − λ (cid:1) + m ξ m ζ (cid:0) λ − κ + 3 λ S (cid:1) + 3 κm ζ (cid:19) + m ξ M (cid:18) − m ξ − m ξ m ζ + 3 m ξ m ζ − m ζ (cid:19) , (34) δd H (cid:3) = − κ
24 + m ξ M (cid:18) m ξ (cid:0) κ − λ − λ S (cid:1) − κm ζ (cid:19) + m ξ M (cid:18) m ξ − m ξ m ζ + 11 m ζ (cid:19) . (35)The terms of Eqs. (34) and (35) can be written in terms of c H , c H (cid:3) along with m ζ , λ S and κ , ( m ξ can be written in terms of these parameters). The one-loop shift terms fromcanonically normalizing the Higgs kinetic energy are, δd shiftH = 3 c H (cid:3) c H ,δd shiftH (cid:3) = 2( c H (cid:3) ) . (36)The one-loop shift terms are O ( v /M ) and can be neglected, since we consistently work tolinear order in the coefficient functions.After performing the one-loop matching at M , the renormalization group is used to evolvethe coefficients to M Z , where the resulting coefficients can be compared with data. Thecomplete set of one-loop anomalous dimension matrices can be found in Refs. [67–69]. Theinclusion of the one-loop matching makes a relatively minor difference in the evolution of We drop the c HD term in Eq. (31) since it doesn’t occur in the singlet model. A more consistent approach would employ the 2-loop anomalous dimensions, however, these are notavailable for the SMEFT.
00 1000 1500 2000 µ R [GeV] C ( t r ee m a t c h i ng ) / C ( - l oop m a t c h i ng ]- C H ❐ C H SMEFT Limit of Singlet Model M H =2 TeV, cos θ =.99, κ =-.5, m ζ =500 GeV, λ S =.03 FIG. 4: Renormalization group evolution of coefficient functions from the matching scale, M , to µ R when the matching is done at tree level and at one-loop for coefficients that are generated attree level. The coefficients are evaluated as a function of the running scale, µ R . C H and C H (cid:3) , as seen in Fig. 4 where we evolve from 2 TeV (note that M H is related to M by Eq. (4)). In Fig. 5, we show the effect of the one-loop matching on the evolution of C HD .In this case, since C HD is zero at tree level, the contributions from the 1 − loop matchingand the renormalization group running are of the same order of magnitude and the effectsare more significant. In Fig. 6 we show the relative size of the 1 − loop matching comparedto the tree level matching as the matching scale M is increased and the overall effects arebetween 10 − C H (cid:3) and C H increase dramatically as thematching scale rises over a few TeV. This is due to the logarithmic running becoming largeand in the case of C H , the 1-loop matching terms become of the same order as the tree levelterms, implying that the perturbative expansion is no longer valid. B. Global Fit
Following Ref. [28], we perform a global fit to the parameters of the non- Z symmetricsinglet model. At the matching scale, M , only the tree level coefficients c H and c H (cid:3) arenon-zero and other coefficients are generated at M Z from the renormalization group running.With tree level matching, the results can be expressed in terms of c H ( M ) and c H (cid:3) ( M ).Using the 1-loop matching at M described in the previous section, additional coefficients14
00 1000 1500 2000 µ R [GeV] -0.004-0.00200.0020.0040.0060.0080.01 C HD Tree matching1-loop matchingSMEFT Limit of Singlet Model M H =2 TeV, cos θ =.99, κ =-.5, m ζ =500 GeV, λ S =.03 FIG. 5: Renormalization group evolution of the coefficient function from the matching scale, M ,to µ R for C HD , which is generated only by the renormalization group running in the singlet model. C HD is evaluated as a function of the running scale, µ R . Matching Scale = M [GeV] -0.6-0.4-0.200.2 C ( M Z )[ - l oop m a t c h i ng ] / C ( M Z )[ T r ee m a t c h i ng ] - C H ❐ C HD C H SMEFT Limit of Singlet Model cos θ =.98, κ =.5, m ζ =M/4, λ S =.03 FIG. 6: Shift in the coefficient functions at M Z as a function of the matching scale, M , when thematching is done at tree level and at one-loop. are generated with a distinctive pattern. The 1-loop matching introduces a dependenceon three additional parameter combinations beyond those at generated by the tree levelmatching and we take as our 5 unknown input parameters, M H , sin θ, m ζ , λ S , and κ . Thematching scale, M , is then calculated using Eq. (4). We match the SMEFT coefficients at M and use the 1-loop renormalization group equations to evolve the SMEFT coefficients to The results used to include the effects of C H require | C H | (cid:46) (5 − (cid:0) M/ TeV (cid:1) [70, 71]. Z where we fit to data.The included data are identical to that of Ref. [28] and include Higgs coupling strengthsfrom ATLAS [45], CMS Higgs coupling strengths [46], W + W − , W ± Z , W h and Zh differ-ential measurements including QCD effects as in [72, 73], and precision electroweak mea-surements including QCD and electroweak NLO effects from Table III of Ref. [50]. Wedetermine the 95% confidence level limits using a χ fit, including the new physics effects atlinear order in the SMEFT coefficients.Figs. 7 and 8 contain our major results. In terms of the parameters of the singlet modelgiven in Eq. (3), we fix M H = 2 TeV and determine the maximum allowed value of sin θ in terms of the other unknown parameters of the model, λ S , κ, and m ζ . The curves arerelatively insensitive to m ζ and λ S (RHS), and the major sensitivity is to κ (LHS) of Fig. 7.We show the regions excluded by unitarity bounds and by vacuum stability bounds. Theblack curves include the Higgs, diboson, and EWPO data. For κ (cid:46)
8, the inclusion of the1-loop matching makes very little difference, but as κ becomes large and approaches theunitarity bound, the difference between tree level and 1-loop matching can be of O (10%).We separately show the limits from only EWPO limits in magenta and note that the 1-loopmatching slightly improves the bound on sin θ .Another interesting way to look at the results is to look at the maximum allowed valueof sin θ as a function of the heavy Higgs mass, M H , for fixed values of κ, m ζ and λ S asshown in Fig. 8. We see that including the 1-loop matching changes the bound on sin θ onlymarginally. The effect is larger as κ is increased.Single parameter fits to models with an additional scalar have been presented in Ref. [33]and updated in Ref. [34] using the dictionary of Ref. [66]. Assuming C H (cid:3) = − /M and C H = 0, they find a limit M H >
900 GeV at 2 σ . Our tree level matching result of Ref. [28] isroughly compatible with this bound, although we find that the inclusion of the renormaliza-tion group running of the coefficients (in particular C HD which is generated by renormaliza-tion group running) is numerically significant, so the bounds cannot be directly compared. Fits to the singlet model with 1-loop matching, but no renormalization group running, are given inRef. [35], but the results are not in a form that we can compare with. �������� ������������ ��������� ������ ������ ��������������� ��������� ������������ ��������� ������ FIG. 7: 95% C.L. limits on sin θ as a function of κ (LHS) and λ S (RHS) for fixed M H = 2 TeVand m ζ = 500 GeV. The fits with tree-level matching are shown as dashed curves, with solid curvesshowing the 1-loop result. The black curves show the result of a global fit to Higgs, diboson, andelectroweak precision data, while the pink curves only the electroweak precision observables. Theregions to the right of the curves are excluded by the fits. The grey and blue shaded regions areforbidden by unitarity and electroweak vacuum stability requirements, respectively. ��������� ������ ��������� ������������ ��������� ������ FIG. 8: As in Fig. 7, now showing limits on sin θ as a function of the heavy Higgs mass, M H ,with fixed values of κ , λ S and m ζ /M H . Regions above the curves are excluded. I. CONCLUSIONS
We have re-examined the sensitivity of a global fit to electroweak precision observablesand to Higgs and diboson data on the parameters of a scalar singlet model in both thefull UV complete model and in the low energy approximation where the heavy scalar isintegrated out and the parameters are matched to the dimension-6 SMEFT. In the fullsinglet model, we find equivalent limits on the allowed mixing angle from the completeEWPO fit and from the fit to the oblique parameters when M H is heavy. For the case withthe second Higgs boson much lighter than M Z , the oblique parameter limits are not a goodapproximation to the full fit. When the new scalar is very heavy, we integrate it out andmatch to the dimension-6 SMEFT and then perform the global fit both using tree level and1-loop matching at the high scale and derive limits on the parameters of the singlet theoryfrom the SMEFT fit. We find that the effect on the fit of including the 1-loop matchingis never larger than O (10%) and that the results are quite insensitive to variations in thesinglet Lagrangian parameters other than the portal term, κ .Digital data can be found at https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/singlet_21 . Acknowledgements
SD is supported by the United States Department of Energy under Grant Contract DE-SC0012704. The work of PPG has received financial support from Xunta de Galicia (Centrosingular de investigaci´on de Galicia accreditation 2019-2022), by European Union ERDF,and by “Mar´ıa de Maeztu” Units of Excellence program MDM-2016-0692 and the SpanishResearch State Agency. The work of SH was supported by DOE Grant DE-SC0013607 andby the Alfred P. Sloan Foundation Grant No. G-2019-12504. [1] M. Bowen, Y. Cui, and J. D. Wells, “Narrow trans-TeV Higgs bosons and H → hh decays:Two LHC search paths for a hidden sector Higgs boson,” JHEP (2007) 036, arXiv:hep-ph/0701035 [hep-ph] .
2] D. O’Connell, M. J. Ramsey-Musolf, and M. B. Wise, “Minimal Extension of the StandardModel Scalar Sector,”
Phys. Rev.
D75 (2007) 037701, arXiv:hep-ph/0611014 [hep-ph] .[3] S. Dawson and I. M. Lewis, “NLO corrections to double Higgs boson production in theHiggs singlet model,”
Phys. Rev.
D92 no. 9, (2015) 094023, arXiv:1508.05397 [hep-ph] .[4] M. M¨uhlleitner, M. O. Sampaio, R. Santos, and J. Wittbrodt, “ScannerS: Parameter Scansin Extended Scalar Sectors,” arXiv:2007.02985 [hep-ph] .[5] T. Robens and T. Stefaniak, “Status of the Higgs Singlet Extension of the Standard Modelafter LHC Run 1,”
Eur.Phys.J.
C75 no. 3, (2015) 104, arXiv:1501.02234 [hep-ph] .[6] R. Costa, M. M¨uhlleitner, M. O. P. Sampaio, and R. Santos, “Singlet Extensions of theStandard Model at LHC Run 2: Benchmarks and Comparison with the NMSSM,”
JHEP (2016) 034, arXiv:1512.05355 [hep-ph] .[7] C.-Y. Chen, S. Dawson, and I. M. Lewis, “Exploring resonant di-Higgs boson production inthe Higgs singlet model,” Phys. Rev.
D91 no. 3, (2015) 035015, arXiv:1410.5488[hep-ph] .[8] S. J. Huber, T. Konstandin, T. Prokopec, and M. G. Schmidt, “Electroweak PhaseTransition and Baryogenesis in the nMSSM,”
Nucl. Phys. B (2006) 172–196, arXiv:hep-ph/0606298 .[9] S. Profumo, M. J. Ramsey-Musolf, and G. Shaughnessy, “Singlet Higgs phenomenology andthe electroweak phase transition,”
JHEP (2007) 010, arXiv:0705.2425 [hep-ph] .[10] J. R. Espinosa, T. Konstandin, and F. Riva, “Strong Electroweak Phase Transitions in theStandard Model with a Singlet,” Nucl. Phys.
B854 (2012) 592–630, arXiv:1107.5441[hep-ph] .[11] V. Barger, D. J. H. Chung, A. J. Long, and L.-T. Wang, “Strongly First Order PhaseTransitions Near an Enhanced Discrete Symmetry Point,”
Phys. Lett. B (2012) 1–7, arXiv:1112.5460 [hep-ph] .[12] S. Profumo, M. J. Ramsey-Musolf, C. L. Wainwright, and P. Winslow, “Singlet-catalyzedelectroweak phase transitions and precision Higgs boson studies,”
Phys. Rev.
D91 no. 3,(2015) 035018, arXiv:1407.5342 [hep-ph] .[13] D. Curtin, P. Meade, and C.-T. Yu, “Testing Electroweak Baryogenesis with FutureColliders,”
JHEP (2014) 127, arXiv:1409.0005 [hep-ph] .[14] A. V. Kotwal, M. J. Ramsey-Musolf, J. M. No, and P. Winslow, “Singlet-catalyzed lectroweak phase transitions in the 100 TeV frontier,” Phys. Rev.
D94 no. 3, (2016) 035022, arXiv:1605.06123 [hep-ph] .[15] T. Huang, J. M. No, L. Pernie, M. Ramsey-Musolf, A. Safonov, M. Spannowsky, andP. Winslow, “Resonant di-Higgs boson production in the b ¯ bW W channel: Probing theelectroweak phase transition at the LHC,” Phys. Rev.
D96 no. 3, (2017) 035007, arXiv:1701.04442 [hep-ph] .[16] C.-Y. Chen, J. Kozaczuk, and I. M. Lewis, “Non-resonant Collider Signatures of aSinglet-Driven Electroweak Phase Transition,”
JHEP (2017) 096, arXiv:1704.05844[hep-ph] .[17] G. Kurup and M. Perelstein, “Dynamics of Electroweak Phase Transition In Singlet-ScalarExtension of the Standard Model,” Phys. Rev. D no. 1, (2017) 015036, arXiv:1704.03381 [hep-ph] .[18] H.-L. Li, M. Ramsey-Musolf, and S. Willocq, “Probing a scalar singlet-catalyzed electroweakphase transition with resonant di-Higgs boson production in the 4 b channel,” Phys. Rev.
D100 no. 7, (2019) 075035, arXiv:1906.05289 [hep-ph] .[19] N. Craig, C. Englert, and M. McCullough, “New Probe of Naturalness,”
Phys. Rev. Lett. no. 12, (2013) 121803, arXiv:1305.5251 [hep-ph] .[20] D. Curtin and P. Saraswat, “Towards a No-Lose Theorem for Naturalness,”
Phys. Rev. D no. 5, (2016) 055044, arXiv:1509.04284 [hep-ph] .[21] V. Silveira and A. Zee, “Scalar Phantoms,” Phys. Lett. B (1985) 136–140.[22] J. McDonald, “Gauge singlet scalars as cold dark matter,”
Phys. Rev. D (1994)3637–3649, arXiv:hep-ph/0702143 .[23] C. P. Burgess, M. Pospelov, and T. ter Veldhuis, “The Minimal model of nonbaryonic darkmatter: A Singlet scalar,” Nucl. Phys. B (2001) 709–728, arXiv:hep-ph/0011335 .[24] A. Menon, D. E. Morrissey, and C. E. M. Wagner, “Electroweak baryogenesis and darkmatter in the nMSSM,”
Phys. Rev. D (2004) 035005, arXiv:hep-ph/0404184 .[25] X.-G. He, T. Li, X.-Q. Li, J. Tandean, and H.-C. Tsai, “Constraints on Scalar Dark Matterfrom Direct Experimental Searches,” Phys. Rev. D (2009) 023521, arXiv:0811.0658[hep-ph] .[26] M. Gonderinger, Y. Li, H. Patel, and M. J. Ramsey-Musolf, “Vacuum Stability,Perturbativity, and Scalar Singlet Dark Matter,” JHEP (2010) 053, arXiv:0910.3167 hep-ph] .[27] Y. Mambrini, “Higgs searches and singlet scalar dark matter: Combined constraints fromXENON 100 and the LHC,” Phys. Rev. D (2011) 115017, arXiv:1108.0671 [hep-ph] .[28] S. Dawson, S. Homiller, and S. D. Lane, “Putting standard model EFT fits to work,” Phys.Rev. D no. 5, (2020) 055012, arXiv:2007.01296 [hep-ph] .[29] J. Brehmer, A. Freitas, D. Lopez-Val, and T. Plehn, “Pushing Higgs Effective Theory to itsLimits,”
Phys. Rev.
D93 no. 7, (2016) 075014, arXiv:1510.03443 [hep-ph] .[30] B. Henning, X. Lu, and H. Murayama, “What do precision Higgs measurements buy us?,” arXiv:1404.1058 [hep-ph] .[31] B. Henning, X. Lu, and H. Murayama, “How to use the Standard Model effective fieldtheory,”
JHEP (2016) 023, arXiv:1412.1837 [hep-ph] .[32] M. Gorbahn, J. M. No, and V. Sanz, “Benchmarks for Higgs Effective Theory: ExtendedHiggs Sectors,” JHEP (2015) 036, arXiv:1502.07352 [hep-ph] .[33] J. Ellis, C. W. Murphy, V. Sanz, and T. You, “Updated Global SMEFT Fit to Higgs,Diboson and Electroweak Data,” JHEP (2018) 146, arXiv:1803.03252 [hep-ph] .[34] J. Ellis, M. Madigan, K. Mimasu, V. Sanz, and T. You, “Top, Higgs, Diboson andElectroweak Fit to the Standard Model Effective Field Theory,” arXiv:2012.02779[hep-ph] .[35] Anisha, S. Das Bakshi, J. Chakrabortty, and S. K. Patra, “A Step Toward ModelComparison: Connecting Electroweak-Scale Observables to BSM through EFT and BayesianStatistics,” arXiv:2010.04088 [hep-ph] .[36] D. Egana-Ugrinovic and S. Thomas, “Effective Theory of Higgs Sector Vacuum States,” arXiv:1512.00144 [hep-ph] .[37] S. Das Bakshi, J. Chakrabortty, and M. Spannowsky, “Classifying Standard ModelExtensions Effectively with Precision Observables,” arXiv:2012.03839 [hep-ph] .[38] G. D. Kribs, A. Maier, H. Rzehak, M. Spannowsky, and P. Waite, “Electroweak obliqueparameters as a probe of the trilinear Higgs boson self-interaction,” Phys. Rev. D no. 9,(2017) 093004, arXiv:1702.07678 [hep-ph] .[39] A. Falkowski, C. Gross, and O. Lebedev, “A second Higgs from the Higgs portal,” JHEP (2015) 057, arXiv:1502.01361 [hep-ph] .[40] M. Jiang, N. Craig, Y.-Y. Li, and D. Sutherland, “Complete One-Loop Matching for a inglet Scalar in the Standard Model EFT,” JHEP (2019) 031, arXiv:1811.08878[hep-ph] . [Erratum: JHEP 01, 135 (2021)].[41] U. Haisch, M. Ruhdorfer, E. Salvioni, E. Venturini, and A. Weiler, “Singlet night inFeynman-ville: one-loop matching of a real scalar,” JHEP (2020) 164, arXiv:2003.05936 [hep-ph] . [Erratum: JHEP 07, 066 (2020)].[42] T. Cohen, X. Lu, and Z. Zhang, “Functional Prescription for EFT Matching,” arXiv:2011.02484 [hep-ph] .[43] G. Buchalla, O. Cata, A. Celis, and C. Krause, “Standard Model Extended by a HeavySinglet: Linear vs. Nonlinear EFT,” Nucl. Phys.
B917 (2017) 209–233, arXiv:1608.03564[hep-ph] .[44] S. Dawson, A. Ismail, and I. Low, “What’s in the loop? The anatomy of double Higgsproduction,”
Phys. Rev. D no. 11, (2015) 115008, arXiv:1504.05596 [hep-ph] .[45] ATLAS
Collaboration, G. Aad et al. , “Combined measurements of Higgs boson productionand decay using up to 80 fb − of proton-proton collision data at √ s = 13 TeV collected withthe ATLAS experiment,” Phys. Rev. D no. 1, (2020) 012002, arXiv:1909.02845[hep-ex] .[46]
CMS
Collaboration, “Combined Higgs boson production and decay measurements with upto 137 fb − of proton-proton collision data at √ s = 13 TeV,”. https://cds.cern.ch/record/2706103 .[47] ATLAS
Collaboration, G. Aad et al. , “Combination of searches for Higgs boson pairs in pp collisions at √ s =13 TeV with the ATLAS detector,” Phys. Lett. B (2020) 135103, arXiv:1906.02025 [hep-ex] .[48] W. F. L. Hollik, “Radiative Corrections in the Standard Model and their Role for PrecisionTests of the Electroweak Theory,”
Fortsch. Phys. (1990) 165–260.[49] A. Freitas, “Higher-order electroweak corrections to the partial widths and branching ratiosof the Z boson,” JHEP (2014) 070, arXiv:1401.2447 [hep-ph] .[50] S. Dawson and P. P. Giardino, “Electroweak and QCD corrections to Z and W poleobservables in the standard model EFT,” Phys. Rev. D no. 1, (2020) 013001, arXiv:1909.02000 [hep-ph] .[51] S. Carrazza, R. K. Ellis, and G. Zanderighi, “QCDLoop: a comprehensive framework forone-loop scalar integrals,”
Comput. Phys. Commun. (2016) 134–143, rXiv:1605.03181 [hep-ph] .[52] D. Lopez-Val and T. Robens, “∆ r and the W-boson mass in the singlet extension of thestandard model,” Phys. Rev.
D90 no. 11, (2014) 114018, arXiv:1406.1043 [hep-ph] .[53] S. Dawson and W. Yan, “Hiding the Higgs Boson with Multiple Scalars,”
Phys. Rev.
D79 (2009) 095002, arXiv:0904.2005 [hep-ph] .[54] C. Englert, J. Jaeckel, M. Spannowsky, and P. Stylianou, “Power meets Precision to explorethe Symmetric Higgs Portal,”
Phys. Lett. B (2020) 135526, arXiv:2002.07823[hep-ph] .[55] M. E. Peskin and T. Takeuchi, “Estimation of oblique electroweak corrections,”
Phys. Rev.D (1992) 381–409.[56] Particle Data Group
Collaboration, P. Zyla et al. , “Review of Particle Physics,”
PTEP no. 8, (2020) 083C01.[57] A. Ilnicka, T. Robens, and T. Stefaniak, “Constraining Extended Scalar Sectors at the LHCand beyond,”
Mod. Phys. Lett.
A33 no. 10n11, (2018) 1830007, arXiv:1803.03594[hep-ph] .[58] G. Chalons, D. Lopez-Val, T. Robens, and T. Stefaniak, “The Higgs singlet extension atLHC Run 2,”
PoS
DIS2016 (2016) 113, arXiv:1606.07793 [hep-ph] .[59] J. de Blas, M. Ciuchini, E. Franco, S. Mishima, M. Pierini, L. Reina, and L. Silvestrini, “TheGlobal Electroweak and Higgs Fits in the LHC era,”
PoS
EPS-HEP2017 (2017) 467, arXiv:1710.05402 [hep-ph] .[60] T. Robens and T. Stefaniak, “LHC Benchmark Scenarios for the Real Higgs SingletExtension of the Standard Model,”
Eur. Phys. J. C no. 5, (2016) 268, arXiv:1601.07880[hep-ph] .[61] B. W. Lee, C. Quigg, and H. Thacker, “The Strength of Weak Interactions at VeryHigh-Energies and the Higgs Boson Mass,” Phys. Rev. Lett. (1977) 883–885.[62] B. W. Lee, C. Quigg, and H. Thacker, “Weak Interactions at Very High-Energies: The Roleof the Higgs Boson Mass,” Phys. Rev. D (1977) 1519.[63] S. Dawson and C. W. Murphy, “Standard Model EFT and Extended Scalar Sectors,” Phys.Rev.
D96 no. 1, (2017) 015041, arXiv:1704.07851 [hep-ph] .[64] W. Buchmuller and D. Wyler, “Effective Lagrangian Analysis of New Interactions andFlavor Conservation,”
Nucl. Phys. B (1986) 621–653.
65] A. Dedes, W. Materkowska, M. Paraskevas, J. Rosiek, and K. Suxho, “Feynman rules for theStandard Model Effective Field Theory in R ξ -gauges,” JHEP (2017) 143, arXiv:1704.03888 [hep-ph] .[66] J. de Blas, J. C. Criado, M. Perez-Victoria, and J. Santiago, “Effective description of generalextensions of the Standard Model: the complete tree-level dictionary,” JHEP (2018) 109, arXiv:1711.10391 [hep-ph] .[67] E. E. Jenkins, A. V. Manohar, and M. Trott, “Renormalization Group Evolution of theStandard Model Dimension Six Operators I: Formalism and lambda Dependence,” JHEP (2013) 087, arXiv:1308.2627 [hep-ph] .[68] E. E. Jenkins, A. V. Manohar, and M. Trott, “Renormalization Group Evolution of theStandard Model Dimension Six Operators II: Yukawa Dependence,” JHEP (2014) 035, arXiv:1310.4838 [hep-ph] .[69] R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott, “Renormalization Group Evolutionof the Standard Model Dimension Six Operators III: Gauge Coupling Dependence andPhenomenology,” JHEP (2014) 159, arXiv:1312.2014 [hep-ph] .[70] G. Degrassi, M. Fedele, and P. P. Giardino, “Constraints on the trilinear Higgs self couplingfrom precision observables,” JHEP (2017) 155, arXiv:1702.01737 [hep-ph] .[71] G. Degrassi, P. P. Giardino, F. Maltoni, and D. Pagani, “Probing the Higgs self coupling viasingle Higgs production at the LHC,” JHEP (2016) 080, arXiv:1607.04251 [hep-ph] .[72] J. Baglio, S. Dawson, and S. Homiller, “QCD corrections in Standard Model EFT fits to W Z and
W W production,”
Phys. Rev. D no. 11, (2019) 113010, arXiv:1909.11576[hep-ph] .[73] J. Baglio, S. Dawson, S. Homiller, S. D. Lane, and I. M. Lewis, “Validity of standard modelEFT studies of VH and VV production at NLO,”
Phys. Rev. D no. 11, (2020) 115004, arXiv:2003.07862 [hep-ph] ..