Probing 6D operators at future e−e+ colliders

Journal of High Energy Physics, May 2018

Abstract We explore the sensitivities at future e−e+ colliders to probe a set of six-dimensional operators which can modify the SM predictions on Higgs physics and electroweak precision measurements. We consider the case in which the operators are turned on simultaneously. Such an analysis yields a “conservative” interpretation on the collider sensitivities, complementary to the “optimistic” scenario where the operators are individually probed. After a detail analysis at CEPC in both “conservative” and “optimistic” scenarios, we also considered the sensitivities for FCC-ee and ILC. As an illustration of the potential of constraining new physics models, we applied sensitivity analysis to two benchmarks: holographic composite Higgs model and littlest Higgs model.

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Probing 6D operators at future e−e+ colliders

HJE e+ Wen Han Chiu 0 2 3 5 Sze Ching Leung 0 1 2 5 Tao Liu 0 2 5 Kun-Feng Lyu 0 2 5 Lian-Tao Wang 0 2 3 4 Clear Water Bay 0 2 Kowloon 0 2 Hong Kong S.A.R. 0 2 P.R.C. 0 2 0 Pittsburgh , PA 15260 , U.S.A 1 Department of Physics and Astronomy, University of Pittsburgh 2 Chicago , Illinois 60637 , U.S.A 3 Enrico Fermi Institute, University of Chicago , Chicago 4 Kavli Institute for Cosmological Physics, University of Chicago 5 Department of Physics, The Hong Kong University of Science and Technology We explore the sensitivities at future e e+ colliders to probe a set of sixdimensional operators which can modify the SM predictions on Higgs physics and electroweak precision measurements. We consider the case in which the operators are turned on simultaneously. Such an analysis yields a conservative" interpretation on the collider sensitivities, complementary to the optimistic" scenario where the operators are individually probed. After a detail analysis at CEPC in both \conservative" and \optimistic" scenarios, we also considered the sensitivities for FCC-ee and ILC. As an illustration of the potential of constraining new physics models, we applied sensitivity analysis to two benchmarks: holographic composite Higgs model and littlest Higgs model. Beyond Standard Model; Higgs Physics - 1 Introduction 2 Analysis formalism 3 Observables for analysis 3.1 3.2 3.3 4.1 4.2 Higgs events Higgs production angular observables Electroweak precision tests 4 Analysis of sensitivity to new physics CEPC analysis: turning on operators individually CEPC analysis: turning on multiple operators simultaneously 4.3 Comparative study at future e e+ colliders 5 Application to two benchmark composite Higgs models 6 Conclusions A Feynman rules for the interaction vertices B Observables for analysis: numerical formulae C Normalized correlation matrices D Parameter marginalization in 2 E 2D 2 analysis the electroweak (EW) scale can be parametrized by a set of six-dimensional (6D) operators Le = LSM + X c i 2 Oi : i (1.1) Here LSM describes physics in the SM. ci and denote dimensionless Wilson coe cients and the cuto scale de ned by the BSM physics, respectively. Among these operators, 59 are CP-even and 17 are CP-odd. The form of the operators depends on the choice of basis [8{13]. Since the discovery of Higgs boson, the probe of the 6D operators, particularly the ones HJEP05(218) motivated by Higgs physics, at LHC and future e e+ colliders has been extensively studied [14{24]. There are di erent strategies in analyzing the sensitivities to new physics. It can be done with only a single operator tuning on at a time, which provides an \optimistic" projection of the sensitivities at the future e e+ colliders. However, new physics models tend to generate multiple such operators. Without assuming a particular model, one could go to the other extreme by turning on all operators simultaneously without assuming any correlation among them. Such an analysis, a primary e ort in this paper, will result in a \conservative" interpretation on collider sensitivities due to cancellation e ects among the multiple contributions. Despite this, we should keep in mind that while this approach give some information about potential degeneracies and correlations in interpreting the measurements, it is not directly applicable to speci c models. New physics models typically generate a smaller set of independent operators, equivalently, predicts correlations between di erent operators in the complete set. For that case, one can analyze the experimental constraints or the collider sensitivities straightforwardly, utilizing the correlation matrix predicted by the speci c models. It is not necessary (and also impossible) to go through all potential new physics models, for the purpose of qualitatively demonstrating the capability of a future collider. As an illustration, we pursued such analyses in two benchmark models: the holographic composite Higgs model and littlest Higgs model. Our study partially overlaps with some recent studies on the sensitivities of probing the SM EFT at future e e+ colliders [21, 25{27]. The study in ref. [21] was pursued under a yet-to-be-explicitly-established assumption that the 6D EW operators can be constrained su ciently well. Di erent from that, we incorporate the sensitivity analysis for these 6D EW operators, without making any rst working assumption about them. This may yield a signi cant impact for the sensitivity discussions on the triple gauge coupling (TGC) measurement. In addition, a recently proposed operating scenario (see, e.g., [28]) is assumed for the FCC-ee analysis. Refs. [25, 26] took similar strategies, with the results presented in the \ "-scheme and in the 6D operator-scheme, respectively. Compared to these analysis, we focus more on the comparative studies on the sensitivities in the \optimistic" and \conservative" scenarios, and the sensitivities at the CEPC, ILC and FCC-ee. More than that, there exist some di erences between the operator sets studied and the observables applied. We include the operator OL(3L)l (as is de ned in table 1) in the analysis which was ignored in [26]. But, unlike [26] (and also [21]), our analysis does not include the Higgs decay observables, and correspondingly several operators which are sensitive to them. As O6 = jHyHj3 OL(3L)l = (LL aLL)(LL $ $ OLl = (iHyD H)(LL LL) ORe = (iHyD H)(lR lR) aLL) aLL) We organize this article in the following way. We will introduce the analysis formalism and the observables applied in section 2 ad section 3, respectively. The analysis and its results will be presented in section 4. In this section, we will pursue a 2 t on the sensitivities of probing the 6D operators at CEPC, in both \optimistic" and \conservative" interpretations. Then we will make a comparative study on the sensitivities at CEPC, FCCee, ILC250 (with data at 250 GeV and below) and ILC (with full data), and look into the operators O6 in details which is di cult to probe. We will apply the analysis to study the theory of SILH in section 5, analyzing the collider sensitivities to probe its benchmarks: holographic composite Higgs model [29, 30] and littlest Higgs model [31]. We conclude in section 6. More technical details and analysis results can be found in appendix. 2 Analysis formalism There are 13 6D operators which are relevant to the e e+ ! ZH production: 10 CP-even and 3 CP-odd ones. In this article, we focus only on the CP-even ones. We also include the triple gauge boson operator since it is often generated together with these ones in new physics scenarios. These 11 operators are summarized in table 1. This is a subset of the operators in the so called Warsaw basis [9], omitting operators with quarks. These 11 operators can in uence physics at the EW scale in four ways: (1) renormalizing wave function; (2) shifting the de nition of EW parameters; (3) modifying the existing SM couplings (including the charge shifting in the gauge boson currents) and (4) inducing new vertices. c2BB2 g02v2B the kinetic terms of the gauge or Higgs elds. First, we note that c2W W2 g2v2W a We begin with wave-function renormalization. OW W , OW B, OBB and OH will modify W a and B can be absorbed into a rede nition of SM electorweak gauge couplings. With this, the canonically normalized SM gauge and Higgs elds are v 2 2 2 cH v 2 h = Zhh0 = 1 h0 W = ZW W 0 = W 0 Z = ZAA0 + ZX Z0 = 1 v 2 2 cwswgg0cW B A0 v 2 2 (c2w s2w)gg0cW BZ0 { 3 { (2.1) HJEP05(218) mZ m(Zr) ZZ + with ZZ = ZZ 1 and ZA = ZA 1. Here the superscripts \sm" represents the SM de nition, and \(r)" represents the reference or the measured central value used as input for the t. Then the parameter shifts can be denoted as msZm = m(Zr) 1 + GsFm = G(r) 1 + F sm = (r) 1 + (r) ; GF G(r) F with mm(ZrZ) = cT v2 This formalism is independent of the de nition of the eld renormalization factors ZZ and ZA. Hence, in addition to a ect the observable directly, D6 operators can also contribution to the deviation from SM prediction by shifting the de nition of input parameters. From here on, we will suppress the superscript (r) for the measured observables, unless speci ed. Since vs2m= 2 di ers with v2= 2 only at O( v44 ) order, we also replace the former with the latter. The new physics corrections to some observables can be derived directly. Here g, g0 are the SU(2) and U(1) gauge couplings and cw and sw are the cosine and sine of the Weinberg angle. Zh;W;Z;A are the rescaling factors. OW W and OBB operators can be probed only via the newly introduced vertices like hZ Z . Similarly, though it does not result in a renormalization of the Higgs eld, the operator O6 can modify the Higgs potential, yielding a shift in the Higgs VEV and mass. Such a shift can be absorbed by the de nition of the Fermi constant. The e ect of O6 can be probed only via its contribution to the cubic and quartic Higgs coupling. Three input parameters of the EW sector in the SM, typically chosen to be f ; mZ ; GF g, receive shifts induced by the 6D operators HJEP05(218) GsFm = G(r) msZm = m(Zr) 1 ZZ + sm = (r)(1 2 ZA) 2 cT vs2m One example is We nd Another example is ) s2w = sin 2 w = s2w = s2w 1 2 w = { 4 { (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) e + e− h e + e− Z γ h e + e− We have gZ = Both of them receive linear corrections from OW B, OT , OL(3L)l and OL(3)l. 3 Observables for analysis Throughout this paper, we will consider three classes of observables: inclusive signal rates of Higgs events, angular observables in Higgs events, and electroweak precision observables (EWPOs). We will not include the total width of Higgs boson and its decay branching ratios. Correspondingly, we will not consider the operators which do not enter the inclusive production rates at tree level, but modify the Higgs decays, such as h ! bb; , only. The incorporation of the Higgs decays as observables could reveal more information about a larger set of operators. We will leave such an important analysis to a future study. Regarding theoretical predictions, we will use \ " to denote the shift caused by wave function renormalization or by de nition shift in the EW input parameters. We will use \ " to denote the total deviation from the reference value for any given observables. 3.1 Higgs events A. Higgs strahlung process. The rst important process is e+e in gure 1. The signal events can be well-selected using the variable of recoiling mass. At leading order, the relevant Lagrangian is given by ! Zh, as is shown LZh v 2m2Z (1 + c(Z1Z) )hZ Z + c(Z2Z) hZ Z + cAZ hZ A + gL(1)Z eL eL (3.1) +gR(1)Z eR eR + gL(2)hZ eL eL + gR(2)hZ eR eR + eA (eL eL + eR eR) ; (2.8) HJEP05(218) with the coe cients c(Z1Z) = c(Z2Z) = cAZ = 2v 2v 2 1 GF + cwswg02cBB { 5 { In this Lagrangian, new vertices appear due to OLl, OL also give rise to a term with new Lorentz structure hZ (3)l and ORe. OW W , OW B and OBB Z . Both yield extra contributions to the production e+e ! Zh, as is indicated in gure 1. B. W W fusion process. Another important process is the W W fusion Higgs production e eh, as shown in gure 2. Here we didn't take into account the Z associated Higgs production, with the Z boson decaying into two neutrinos. At leading order, the relevant Lagrangian is given by (1 + c(W1))hW +W + c(W2)hW + W + p (1 + c(W3))(W + L eL + W eL L) + c(W4)(hW + L eL + hW eL g2v 2 g 2 + c(5)(hZ with the coe cients c(W1) = c(W3) = gZ gZ gZ gZ sw sw cw cw L) (3.3) (3.4) L w w + L 2GF c(3)lv2 2 L + hZ L L) ; GF + Zh c(W2) = 2cW W g2v 2 c(W4) = p c(L3)lgv The Wilson coe cients of OH , OT and OLL rescaling of the SM couplings. OW W , OLl and OL (3)l yield two new vertices. (3)l only appear in c(W1) and c(W3), resulting in a C. Z-associated di-higgs process. As the beam energy increases, di-Higgs channel switches on. An important channel is the Z association production process e e+ ! Zhh. { 6 { h h e + e− e + e− h Z/γ Z h h h Z h e + e− h e + e− Z/γ h Z/γ Z h h h The relevant Lagrangian for this channel is LZhh LZh + (1 + c(Z3Z) )hhZ Z + c(Z4Z) hhZ + gL(3)hhZ eL eL + gR(3)hhZ eR with the coe cients c(Z3Z) = L g(3) = 3 = GF + GF mZ gZ clL + c(3)l L 2 2 mZ + 2 ZZ + 2 Zh Higgs production angular observables A recent discussion on the angular observables for the process e e+ be found in [32, 33]. Among the six independent angular observables, four are CP-even, ! hZ(! l+l ) can given by d d cos 1 Z 1 1 { 7 { (3.5) (3.6) (3.7) Here the angular variables are de ned as in gure 4. The EWPOs at Z pole which are relevant to our analysis include At tree level, the Z partial decay width and the asymmetry are given by f = NC 12 f m(Zr) uv ut1 Af = 2gVf gVf + gAf f = (Z ! f f ) = NC 12 f m(Zr) uv ut1 Af = g g 2 L g 2 R L2 + gR2 in terms of vector and axial couplings gVf;A, or by 4mf2 " m(Zr)2 jgVf j2 + jgAfj2 + m2 m(Zr)f22 (jgVf j2 2jgAfj2) # 4mf2 " 1 m(Zr)2 2 (gL2 + gR2) + 2mf2 m(Zr)2 g 2 L 4 g 2 R 4 3 2 gLgR # { 8 { (3.8) (3.9) (3.10) (3.11) in terms of chiral couplings gL;R. l; is de ned for a single avor, whereas inv includes contribution from all possible avors. With the 6D operators turned on, the corrections to the chiral couplings of Z boson are given by Charged lepton gZ + gZ 8swcw 4s2w 3 2cw sw gL = gZ gZ gZ gZ + gZ 4swcw 2s2w 3 2cw sw w represent the e ect of the EW parameter shift; ZZ and ZX represent the e ect of eld rede nition; and c(L3)l, clL and ceR represent the e ect of the charge shift in the leptonic Z current. The quark current operators are turned o in this paper, though they may contribute to some of these observables, e.g., Rb, in a more general context. For more discussions on this, see, e.g., [26]. The formulae for the operator corrections to the EWPOs are presented in appendix B, with six Wilson coe cients involved: cW B, cT , c(L3)l, cLL (3)l, clL and ce . As is indicated in R eq. (B.1){(B.9), the ratio for the coe cients of cW B, cT and c(L3L)l in the EWPOs, N , Ab, AFB, AbFB, Rb, R , R and sin2 elep, is xed to be terms in these EWPOs are generated either via charged leptons, up quarks and down quarks, or via 1:1 : 2 : 4. This is because the three gLi=gLi gL=gL gRi=gRi, with i representing gRl=gRl. Both of them satisfy the relation g i L g i L g i g i R ; R g L g L g l R g l R s2w w ZX + 2 w ZX + (3.15) with the combination 2 w ZX xing this ratio. This combination also contains a c(L3)l term with its coe cient having a xed ratio with the other ones, 1:1 : 2 : 4 : 4. However, this ratio does not hold in AFB, AbFB, R , R e and sin2 lep due to extra contributions proportional to c(L3)l + clL. Neither does it hold in N due to both c(L3)l clL which are caused by the charge shift in the Z boson current. The charge shift can receive contributions from ORe as well. So the set of EWPOs at Z pole depend on four of the six Wilson coe cients or their linear combinations: 0 = 1:1cW B + 2cT 4c(L3)l + 4c(L3L)l, = c(L3)l clL and ce , R leaving at least two degenerate or approximately degenerate directions. More explicitly, we have N . It depends on 0, and ceR. Ab and Rb. They only depend on 0. e gZ { 9 { MZ(GeV) GF (10 10GeV 2) | | mt[GeV](pole) AbF;B and sin2 elep. They have the same dependence on 0 , + and ceR. R ; . They have the same dependence on 0 , + and ceR. These degenerate or approximately degenerate directions could be lifted by Z , which is approximately proportional to gLigLi + gi gi , and mW . R R Z and mW have di erent dependences on the variables beyond 0; and ce . Thus, we have totally six classes of R non-degenerate EWPOs to probe the six Wilson coe cients. The entangled dependence of the EWPOs on the six operators also explains the relatively large magnitude for their correlation matrix entries, as are listed in appendix C. Though sin2 lep and s e 2w are identical in the SM, they represent di erent measurements. Hence they are in uenced by these 6D operators in di erent ways. s corrections via the EW parameter shift only (see eq. (2.6)), whereas sin2 lep receives extra 2w received e contributions caused by eld rede nition (see eq. (3.15)). B. W boson mass. The W boson mass mW = mZ cw receives contributions via the shift of the EW parameters only, resulting in MW MW = gZ gZ sw cw w 1 GF : 2 GF (3.16) C. Di-boson process. The di-boson production e e+ ! W +W can be applied to probe the TGC, and hence the operator O3W . It is mainly in uenced by the coupling shift in gZ due to OW B, OT , OL(3L)l and OL (3)l (see eq. (2.8)), and the charge shift in the electron current of Z boson caused by OLl and ORe. Despite this, a full angular analysis might be valuable, given that the total signal rate is dominated by forward transverse W W production and hence less sensitive to anomalous couplings. We leave the latter to a future work. 4 Analysis of sensitivity to new physics Before performing a full analysis on the sensitivities of probing the 6D operators at future e e+ colliders, we will start with a set of analysis using CEPC as an example. We begin with the case in which we turn on one operator at a time. This simpli ed approach provides an optimistic estimation on the energy scales that could be probed. It provides a basic idea on how the 6D operators individually contribute to the observables, but the potential cancellations among the contributions from di erent operators are ignored. The latter could dramatically change the collider sensitivities. To illustrate this point, we will consider several cases with more operators turned on. Finally, we will study the sensitivities at all HJEP05(218) Observables (Zh) ( h) (Zhh) (W+W ) N A A (3) (4) by the input parameter uncertainties which are summarized in table 2. This error is negligibly small for the observables except Z and sin2 elep. HJEP05(218) ILC can be precisely measured via e e+ ! Zh at future colliders. and CEPC. A recently proposed operating scenario (see, e.g., [28]) has been assumed for the FCC-ee analysis. A beam polarization con guration of (Pe ; Pe+ ) = ( 0:8; 0:3) is assumed for ILC at 250 and 500 GeV. The errors presented are all relative, except the ones for MW ; Z and sin2 elep. The subscript \th" and \in" denotes errors caused by theoretical and input parameter uncertainties, respectively. The numbers in red are obtained by rescaling the experimental errors provided in the referred literatures, which are assumed to be statistical-error-like. As for the precision of measuring ( h) and (Zhh), we assume that the relevant Higgs decay branching ratios (such as Br(h ! bb)) future e e+ colliders. For each of these future programs, multiple operating scenarios have been suggested. We will focus on a subset of them in the analysis. The input parameter values, and the current and projected measurement precisions used for the analysis are summarized in table 2, table 3 and table 4, respectively. We will take into account the impact of the input parameter uncertainties for the measurement precisions. This e ect was discussed in [39] and is denoted as an error with a subscript \in" table 3 and table 4. Also, a running coupling (mZ ) in the MS scheme will be used in the analysis. The numerical ( h) (W +W ) N A A (3) (4) All 0.0189 | | 140 0.00797 0.00561 0.0064 4.75 | | | 163 C.L., with the operators turned on individually. The numbers in red denote the best sensitivity which could be achieved using a single observable, whereas the numbers in the last row represent the sensitivity based on a combination of all observables. formulae for the operator corrections to the observables are summarized in appendix B. The e ective operators are implemented using FeynRules and the cross sections are computed using either CalcHEP or MadGraph5 [43{45]. 4.1 CEPC analysis: turning on operators individually The sensitivities for probing the 6D operators at CEPC are presented in table 5, with them turned on individually. Each row of the table shows the sensitivity of an observable production. The angular observables de ned in e e+ and OL in probing the operators, with the last row showing the combination. OW B, OT , OLL (3)l can be well-probed by the EWPOs, because of the EW parameter shift, the eld rede nition and the charge shift in the Z boson current that they caused. OLl and ORe contribute to the charge shift in the Z boson current, and hence can be also probed e e+ very well. O3W contributes to TGC directly, and can be probed by the measurement of ! W +W . Probing the other four operators, OW W , OBB, OH and O6, mainly relies on the measurement of the Higgs observables, such at the signal rate of e e+ ! Zh ! Zh are less sensitive in probing the operators. As shown in the last row, the combination of the observables can sizably improve the sensitivities to fOW B; OT ; OL(3L)l; OL(3)l; OLl; OReg, compared to other operators. This implies that more than one observables are sensitive to each of these operators, as (3)l was advertised in section 3.3. \Optimistic" (with one operator turned on at a time, denoted by \Individual") and \semi-conservative" (with multiple operators (instead of all operators) turned on, defOW B; OT ; OL(3L)l; OL (3)l quently in the middle and bottom panels. noted by \Marginalized") sensitivity interpretations for probing each of the set of 6D operators fOW B; OT ; OLL ; OL(3)l; OLl; OReg, with the EWPOs at CEPC applied. In the top panel, (3)l g are turned on for marginalization. OLl and ORe are incorporated subseators turned on simultaneously) sensitivity projections for probing each of the set of 6D operators at CEPC. 4.2 CEPC analysis: turning on multiple operators simultaneously Next let us turn on more 6D operators in table 1. For a comparison with the results shown in table 6, we need to project the allowed region in the space of Wilson coe cients to the relevant axis, that is, to \marginalize" the irrelevant Wilson coe cients. There is a geometric interpretation regrading this method. The 2 de nes a 10-dimensional ellipsoid in an 11-dimensional space which is expanded by the set of Wilson coe cients. Marginalizing 10 of the 11 Wilson coe cients is equivalent to imposing the conditions @ci = 0, with i running over all of the 10 Wilson Coe cients. It results in a projection of the ellipsoid to the direction de ned by the 11th Wilson coe cient. This method can be also generalized to the case with less Wilson coe cients being marginalized. An introduction to this statistical method is given in appendix D. We start with the set of six operators fOW B; OT ; OL(3L)l; OL(3)l; OLl; OReg which are expected to be constrained by the six classes of EWPOs at tree level (as is discussed in section 3.3). The CEPC sensitivities for probing each of them are presented in gure 5, with the EWPOs applied only, in both the \optimistic" and \semi-conservative" cases. With the rst four operators turned on (top panel), the CEPC sensitivities decrease from dozens of TeV in the \optimistic" case to O(10) TeV. The turning on of the fth opexcept OLL erator OLl doesn't change the results much (middle panel). However, the turning on of the last operator ORe causes a jump of the CEPC sensitivities for probing these operators (3)l. This is related to the fact that Rb (one of the six classes of the EWPOs) is a weak observable in probing 0 . With the sixth operator turned on, the lack of a sixth independent strong EWPOs yields an approximately degenerate direction in the parameter space expanded by the six operators. To break this degeneracy, extra observables (e.g., Bhabha scattering e e+ ! +), need to be introduced. A full analysis for the CEPC sensitivities for probing the whole set of 6D operators is presented in gure 6, with all observables in table 4 applied. The normalized correlation matrix for this 2 t is presented in table 7 of appendix C. We have the following observations on the \marginalization" results: For the set of operators fOW B; OT ; OL(3L)l; OL(3)l; OLl; OReg, the CEPC sensitivities are inherited from the ones presented in gure 5. The energy scale that the CEPC is able to probe decreases from dozens of TeV in the \optimistic" case to TeV or several TeV, except for OL(3L)l. The operator O3W can be weakly probed only via the e e+ ! W +W with the energy scale accessible to the CEPC being decreased from a couple of TeV production, in the \optimistic" case to sub TeV (this feature is also shared by FCC-ee and ILC, as will be shown below). This is a result of the concerted action of (1) the weak dependence of the e e+ production on O3W due to helicity suppression at linear level [46]; and (2) the existence of approximate degeneracy for the set of EW operators to which the e e+ ! W +W production is much more sensitive (see eq. (B.19)). This e ect yields a sensitivity estimation for probing O3W several times weaker than that obtained in [21]. The three operators fOW W ; OBB; OH g contribute to the Higgs events at tree level. The energy scales that the CEPC is able to probe decrease from several TeV/TeV in the \optimistic" case to TeV/sub TeV, with potential cancellation between the operators taken into account. This is related to the fact that there is only one observable at 240 GeV which is highly sensitive to these operators, say, (Zh). Though ( h) and the e e+ ! Zh angular observables play a role in constraining the Wilson coe cients, they are too weak to completely break the remaining degeneracies. Despite this, the sensitivities for probing OW W ; OBB and OW B could be improved by a couple of times by incorporating the Higgs decay measurements. For example, the decay width of the di-photon mode can be shifted by these operators, yielding 2:95 cBB 2 2 2:94 cW B + 2:95 cW W 2 0:0606c(L3)l + 0:0606c(L3L)l : (4.1) As is indicated in [21, 26], including the di-photon decay measurement may push the sensitivities of probing the OW W and OBB operators up to several TeVs (note, fewer or no relevant EW operators were turned on in [21, 26], which may cause an uncertainty for the estimation). The operator O6 contributes to the Higgs events at loop level only. The energy scales that the CEPC is able to probe decrease from sub TeV in the \optimistic" case to < O(0:1) TeV. The 2 t sensitivities can be also projected to a 2D plane expanded by two Wilson coe cients, using a marginalization method, as is shown in gures 11{13 in appendix E. of the set of 6D operators at CEPC, FCC-ee, ILC250, ILC500 and ILC. Here \ILC250" refers to a combination of the ILC data at 250 GeV and the EW precision measurements at LEP (see table 3); \ILC500" refers to a combination of \ILC250" and the ILC data at 500 GeV; and \ILC" refers to a more optimistic operating scenario, with the LEP measurements in \ILC500" replaced by the Giga-Z data. 4.3 Comparative study at future e e+ colliders Next let us make a comparison on the sensitivities of probing the 6D operators at the future e e+ colliders. For each machine, there exist multiple possibilities for its operating scenario. For concreteness, we consider the measurement precisions at CEPC, FCC-ee and ILC with a subset of possible running scenarios, shown in table 4. The \optimistic" and \conservative" sensitivity interpretations at each machine are presented Figiure 7. Both CEPC and FCC-ee are circular e e+ colliders with non-polarized beams. Bene tting from a larger integrated luminosity at Z pole, the sensitivities at FCC-ee are mildly better than the CEPC ones, in both interpretations. The comparison with the sensitivities at ILC250, ILC500 and ILC is more involved. The ILC250 is less capable in probing these operators than both CEPC and FCC-ee, because of its relatively small luminosity at 250 GeV and the lack of data at Z-pole. However, this can be improved signi cantly by the data expected to be collected at a higher beam energy.1 With the data at 500 GeV, the ILC500 performance becomes not much worse than or comparable to the CEPC and FCC-ee ones in the optimistic case. In the conservative case, the ILC500 performance becomes comparable to or even better than the CEPC and FCC-ee ones. This results in a smaller di erence between the two kinds of sensitivity interpretations at ILC, compared with the ones at CEPC and FCC-ee, as is indicated in gure 7. On the other hand, the data at Giga-Z can slightly improve the sensitivities only which could be achieved at ILC500. 1This feature was also noticed in [26], but an explicit comparison with the CEPC and the FCC-ee performances was missing. OBB ILC250 + (W +W ) + (Zh) + (Zhh) + ( h) = ILC500 in the rst column are all measured at 500 GeV. We note that we have oversimpli ed the beam polarization scenario at ILC, assuming a full-time run for the polarization con guration (Pe ; Pe+ ) = ( 0:8; 0:3). Splitting time between di erent polarization con gurations can enhance the power of breaking the operator degeneracies. This e ect has been discussed in [21, 25], yielding an improvement of 20{30% on the reach of the new physics scale in some of the operators. To get a better picture about the roles played by the observables at 500 GeV, in table 6 we present the marginalized tting results for =pci (TeV) in the ILC scenarios, varying from ILC250 to ILC500 by adding one more observable at 500 GeV each time. Compared with that at CEPC and FCC-ee, the degeneracy problem for fcW B; cT ; c(L3)l; c(3)l; clL; ceRg LL at ILC250 is even worse, given the lack of the Z-pole data. This problem can be addressed measurement at ILC500, as is indicated in table 6. to some extent by the e e+ ! W +W (W +W ) does not depend on fOW W ; OBB; OH ; O6g at tree level, but it has relatively strong sensitivities to these EW operators (see eq. (B.21)). With its help, the constraints for these operators are raised to a level compared to the ones at CEPC and FCC-ee. But this also means that the sensitivity to O3W is still weak. A combination of the other three observables at 500 GeV, say, (Zh), (Zhh) and ( h) can help constrain three of fOW W ; OBB; OH ; O6g which are weakly constrained at ILC250. Particularly, the ILC500 has a much better performance in probing O6, compare to CEPC and FCC-ee. This is due to the e e+ ! Zhh production, an observable which is not available at CEPC and FCC-ee. Though it is less important in the \optimistic" analysis, this observable plays a crucial role in breaking the degeneracy related to O6 in the \conservative" scenario. As for CEPC and the FCC-ee , their weakness in probing O6 could be mitigated somewhat by combining with the LHC data for di-Higgs production, e.g., pp ! hh ! bb [47{ 50]. Note, the weak sensitivity to probe O6 below the Zhh thresholds (particularly in the \conservative" scenario) may indicate that the non-linear c6 terms, e.g., the one-loop quadratic term induced by the Higgs self-energy correction [51], need to be incorporated in the analysis. However, this term, even if being turned on, still fails to yield a bound clearly stronger than the perturbative unitarity one set by the hh ! hh scattering, say, j 3j < 5:5 [52]. So, we simply neglect such terms here. Such a comparative study can be also extended to a plane expanded by two Wilson coe cients, as is shown in gure 14 in appendix E. Application to two benchmark composite Higgs models In this section, we will apply our analysis to a couple of benchmark composite Higgs models. If the composite resonances are heavy, their low energy e ects can be captured by a set of correlated EFT operators, named as a \SILH" parametrization [10]. The SILH parametrization contains two characteristic parameters: f , the decay constant of strong dynamics, and g , the strong coupling. Its Lagrangian is given by [10] LSILH = c~H f 2 OH + c~T f 2 OT c~6 f 2 O6 + c~W m2 OW + c~B m2 OB c~HW c~HB + 16 2f 2 OHW + 16 2f 2 OHB + 16 2f 2g2 OBB + c~ g02 3!g2c~3W 16 2m2 O3W : (5.1) = g f de nes a typical composite resonance mass. To begin with, we neglect the loop-level operators listed in the second line, and rewrite the Lagrangian in the minimal operator basis using the relations [12] OW = g2 OB = g02 3 2 OH + 2O6 + 1 2 (Oyu + Oyd + Oyl + h.c.) + 1 (3)l Here Oy u;d;l denotes the 6D Yukawa operators, say, the product of the Yukawa term and The 6D Yukawa operators Oy the HyH, and f runs over all fermions in the SM. These two relations can be further simpli ed to make connection to our analysis. While substituting OW in eq. (5.1), we omit the operator O6, considering its insensitivity to the observables used in the analysis. u;d;l mainly in uence the Higgs Yukawa couplings and hence are less relevant for the inclusive observables applied. The case for OB is somewhat more complicated. The quark current operators may nontrivially contribute to the had. So we will exclude all EWPOs involving the Z hadronic width had below, in order to safely neglect this subtlety. Then under an assumption of 2 = (4 f )2, the relevant Lagrangian LSILH cH 2 OH + cT c(3)l 2 OT + L L2 OLl + c e R e 2 OR (5.2) (5.3) (5.5) terms are given by with cH = (4 ) 2 c~H L c(3)l = (4 ) 2 g42gc~W2 ; ghW W = gmW 1 c~H v 2 2 f 2 : 3g2c~W 2g2 ; cT = (4 ) 2 c~T clL = (4 ) 2 g402gc~2B ; g02c~B 2g2 ; ceR = (4 ) 2 g202gc~2B : (5.4) The SILH can have di erent realizations, which are characterized by the values of ~cis. Though the LHC runs are able to constrain the SILH, the experimental bounds are typically model-dependent. One LHC probe is to measure the Higgs couplings such as In the right panel, the coordinate axes are in the unit of (TeV) 2 . The solid lines in color and the dashed lines represent the contours of g and f in strong dynamics, respectively. The gray region indicates the ranges de ned by f > 0 and 0 < g < 4 . The current LHC runs yield a lower bound f > 600 700 GeV, for c~H = 1 [ 53, 54 ], under the assumption of no mixing e ect with extra scalars. Such a bound could be pushed up to 1:5 TeV at HL-LHC. Another LHC probe is to search for the composite resonances. For example, the current searches for the fermionic top partner via its pair production set up an lower limit for the resonance mass 0:9 1:2 TeV [55{58], and hence yield a constraint g f = m > O(1) TeV. composite Higgs model and littlest Higgs model. Below we will consider two benchmark models: holographic A. Holographic composite Higgs model. The holographic Higgs model [29, 30] is based on a theory over a slice of ADS5 space-time. This space-time, characterized by a constant radius of curvature for its internal space, is compacti ed with two 4D branes as boundaries. By matching the holographic Higgs model with the SILH EFT, one obtains the Wilson coe cients in the Lagrangian eq. (5.2) as [10] c~T = 0 c~H = 1 c~W = c~B 1 : (5.6) This setup yields a coe cient cH = 2 in the Lagrangian eq. (5.3) which depends on both SILH parameters: f and g . As for the other coe cients, all of them are dependent on (g f )2 only and hence are identical up to a constant factor. The sensitivities of probing the holographic composite Higgs model at future e e+ colliders are presented in gure 8. According to the left panel, the parameter region with a small f or/and a weak g is relatively easy to probe. This is because it yields relatively light composite resonances and hence a lower e ective interacting scale. This observation is consistent with what one had in [27], where the \SILH" pattern is essentially the holographic composite Higgs model discussed here, except that several more operators were turned on in [27]. Note, as the strong coupling g approaches 4 , the loop-level operators in eq. (5.2) may not be negligible in the analysis compared to OW and OB. It is straightforward to project the sensitivities to the planes of the Wilson coe cients. For except the left-upper one, the coordinate axes are in the unit of (TeV) 2. The solid lines in color and the dashed lines represent the contours of g and f in strong dynamics, respectively. The gray region indicates the ranges de ned by f > 0 and 0 < g < 4 . the horizontal axis de ned by cH = 2). illustration, the projection at the cH = 2 c(L3)l= 2 plane is shown in the right panel in gure 8. The projections at the other planes are either a single line (the ones with no axis being de ned by cH = 2), or a rescaling of this panel along the c(L3)l= 2 axis (the ones with B. Littlest Higgs model. The littlest Higgs model [31] is a composite Higgs model with collectively symmetry breaking, with a coset group SU(5)=SO(5). By matching the littlest Higgs model with the SILH EFT, one can gure out the Wilson coe cients in the Lagrangian eq. (5.2) as [10] 1 16 1 4 1 2 c~T = c~H = c~W = c~B = 0 : (5.7) This setup yields two vanishing coe cients in the Lagrangian eq. (5.3): clL= 2 and ceR= 2. The other three coe cients cT = 2, c(L3)l= 2 and cH are dependent on f , g , and both of The sensitivities of probing the littlest Higgs model at future e e+ colliders are presented in gure 9. Similar to the holographic composite Higgs model, the parameter region with a small f or/and a weak g will be probed rst (left-upper panel). The sensitivity projections to the planes expanded by cT = 2, c(L3)l= 2 and cH are also presented. Conclusions In this article we presented a systematic study on the sensitivities of probing the UV physics at the future e e+ colliders. The e ect of new physics is parametrized by a set of 6D operators at leading order in its EFT. We turned on eleven of these operators simultaneously, which can be probed by Higgs physics and EW precision measurements. The analysis provides a \conservative" projection on the collider sensitivities, complementary to the \optimistic" projection presented where these 6D operators are turned on individually. Then we made a comparative study on the sensitivities at CEPC, FCC-ee and ILC. Three running scenarios at ILC were considered: \ILC250" (ILC data at 250 GeV + EWPO mea250 and 500 GeV + GigaZ data). As an application, we analyzed two benchmark models in the composite Higgs scenario. Our results can be brie y summarized as following. up to dozens of TeV by measuring the EWPOs, because of their tree-level contributions to the eld rede nition and the coupling and charge shifts in the Z boson current. fOW W ; OBB; OH ; O3W g can be probed up to TeV or several TeVs by measuring the Higgs observables and the e e+ ! W W production, due to their corrections to the Higgs couplings and TGC, respectively. O6 is di cult to probe because it contributes to e e+ ! Zh at loop level only. These features are shared by FCC-ee and ILC250, ILC500, ILC (though the sensitivities to probe O6 can be improved to some extent by measuring the e e+ ! Zhh production at ILC500 and ILC). In the \conservative" analysis where the set of eleven operators are turned on simultaneously, the energy scales that the CEPC and FCC-ee are able to probe decrease to O(1 10)TeV for fOW B; OT ; OL(3L)l; OL(3)l; OLl; OReg. This is mainly due to an approximate degeneracy caused by the weakness of Rb. For fOW W ; OBB; OH ; O3W g, the sensitivities decrease to TeV or sub TeV, and for O6 to < O(0:1) TeV. Bene tting from a larger integrated luminosity at Z pole, the sensitivities at FCCee are mildly better than the CEPC ones, in both \optimistic" and \conservative" projections. An ILC run with ECM = 500 GeV (ILC500) is highly bene cial. Limited by its relatively small luminosity at 250 GeV and the lack of data at Z-pole, ILC250 is less capable in probing these operators. However, this can be adequately compensated by the data at 500 GeV. By combining with the 500 GeV data, the ILC performance is comparable to or better than the CEPC and FCC-ee ones. Moreover, compared to CEPC and FCC-ee, ILC500 performs much better in probing the O6 operator or measuring the cubic Higgs coupling in the \conservative" analysis. This is mainly because the e e+ ! Zhh production, an observable not available at CEPC and FCC-ee [26], can break the degeneracy related to O6. Additionally, the ILC can also bene t from time splitting among di erent polarization con gurations [21, 25]. As an application, the \conservative" analysis is applied to the simpli ed model of SILH, with the mutual dependence of the Wilson coe cients taken into account. The analysis indicates that CEPC, FCC-ee and ILC have a potential to probe its decay constant up to O(1 10)TeV, with the strong coupling varying between 1 Acknowledgments We would like to thank M. Peskin for valuable comments on the draft, and J. Gu, Y. Jiang and Z. Liu for helpful discussions. We would thank C. Grojean et al. for coordinated publication of their related work. We would acknowledges the hospitality of the Jockey Club Institute for Advanced Study at Hong Kong University of Science and Technology, where part of this work was performed. T. Liu and K. Lyu are supported by the Collaborative Research Fund (CRF) under Grant No. HUKST4/CRF/13G. T. Liu is also supported by the General Research Fund (GRF) under Grant No 16312716. Both the CRF and GRF grants are issued by the Research Grants Council of Hong Kong S.A.R.L.-T. Wang is supported by the U.S. Department of Energy under grant No. DE-SC0013642. A Feynman rules for the interaction vertices The modi ed Feynman rules for the interaction vertices are listed as below h h Z, μ Z, ν Z, μ Z, ν k1 k3 k2 k1 k2 k1 k2 h h h h 3i m2h 1 + 3 Zh + GF 2GF 2c6v4 m2h 2 2i cH2 v(k1 k2 + k1 k3 + k2 k3) gz2v 2 2 z g 2 = ig 1 + 2 ZZ + Zh + + v 2 (v2d1 2 gZ + gZ GF 2GF vd2 (k1 k2)d2) + i 2 k1 k2 = ig 1 + 2 ZZ + Zh + gZ gZ + 1 2 (v2d3 (k1 k2)d2) + i d22 k1 k2 (A.1) (A.2) (A.3) Z, ν A, μ Z, ν A, μ A, ν A, μ A, ν k2 k1 k2 k1 k2 h h h h h Z h h Z h = i 2 vd4 v = i 2 1 = i 2 (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) A, μ = i e (A.10) Z, μ = i gL 1 + ZZ + e ZX (A.11) gZ gZ 2s2w w c2w HJEP05(218) Z, μ = i gz sin2 1 + ZZ + e ZX (A.12) gZ + gZ 2cw w sw Here the Higgs and gauge elds have been rescaled to their canonical forms. The relevant coe cients are de ned as d1 = Z g 2 2 1 2 cH + 2cT + gZ4 (c4wcW W + c2ws2wcW B + s4wcBB) d2 = 4gZ2 (c4wcW W + c2ws2wcW B + s4wcBB) d3 = 3gZ2 cT g 2 2Z cH + gZ4 (c4wcW W + c2ws2wcW B + s4wcBB) d4 = 2gZ2 cwsw( 2s2wcBB (c2w s2w)cW B + 2c2wcW W ) d5 = gZ (cL (3)l + clL) d6 = gZ ceR d7 = 4gZ2 c2ws2w(cW W + cBB cW B) (A.13) B Observables for analysis: numerical formulae The formulae for calculating the contributions of the 6D operators to the observables at future e e+ colliders are listed in the following. The formulae are obtained by using MadGraph and CalcHEP, with the model les generated by FeynRule, or by using Mathematica directly. The e ect of renormalization group running from the cuto to the Z pole or the beam energy scales has been neglected for the Wilson coe cients. In the following, we use the simpli ed notation ci2 1. EWPOs. N Ab A F B AbF B Rb R R N N = 0:00585 cW B 2 cW B 2 0:250 L2 + 0:113 R 2 0:00781 + 0:0142 0:0107 cT e Ab = Ab A AF B F B = cW B 2 cl 0:101 + 0:184 + 0:128 L2 + 0:146 R 2 AbF B = Ab F B cW B 2 cl 0:625 + 1:14 + 0:784 L2 + 0:894 R 2 2 + 0:0139 L 2 (3)l 0:0213 cLL (3)l 2 cT 2 cT 2 e cT 2 c e 0:0285 L (3)l 2 + 0:0285 cLL (3)l 2 0:241 L (3)l 2 + 0:369 cLL (3)l 2 1:50 L c (3)l 2 + 2:28 cLL (3)l 2 Rb Rb = 0:00189 cW B 2 0:00345 cT 2 + 0:00691 L 2 c (3)l 0:00691 cLL (3)l 2 R R R R = 0:00969 + 0:0177 + 0:0353 cLL (3)l 2 0:124 L2 + 0:109 R 2 = 0:00970 + 0:0177 + 0:0354 cLL (3)l 2 0:124 L2 + 0:109 R 2 cl cl cT 2 cT 2 0:159 L c (3)l 2 0:160 L c (3)l 2 c e c e cW B 2 cW B 2 (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) 2. Total signal rates. e+e ! Zh (1) Unpolarized 240 GeV s 2 s2 w w = 0:0483 cW B 2 0:0612 L 0:0881 cT 0:0698 R e 2 2 + 0:115 L 2 c(3)l 0:176 cLL (3)l 2 Z Z = 0:0112 2 cW B + 0:079 cT 0:121 L + 0:158 cLL (3)l 2 0:0113 L 0:0113 R mW mW 0:0111 2 cW B + 0:0433 cT 2 0:0264 L c(3)l 2 + 0:0264 cLL (3)l 2 2 2 2 2 2 2 0:0858 R c e 2 0:343 R c e 2 2 (3)l 2 2 2 c(3)l e 2 2 2 2 2 cW W = 0:225 + 0:0554 cW B + 0:0164 cBB 0:0500 cT 0:0606 cH 2 + 0:627 L cl + 0:264 cLL + 0:891 L 2 0:781 R c 0:00106 (B.12) 0:0613 cW B 2 0:0263 cBB + 0:0779 cT 0:0606 + 1:12 L (3)l cl + 0:520 cLL + 1:64 L 0:000944 (B.13) 0:0617 (3)l cl + 0:520 cLL + 6:54 L 0:0000260 (B.14) (2) Polarized ( 0:8; 0:3) 250 GeV (3) Polarized ( 0:8; 0:3) 500 GeV 0 0 0 = 0:379 = 0:666 2 c(3)l 2 cW W 2 c(3)l 2 cW W 2 c(3)l 2 e+e ! e eh (240 GeV) 0 = 0:0132 0:730 L c(3)l 2 (3)l 2 cl + 0:577 cLL + 0:0136 L 2 cH 2 c6 2 0:000399 (B.15) c6 2 c6 2 c6 2 cH 2 cH 2 (B.9) (B.10) (B.11) (1) Unpolarized 240 GeV = 0:0192 0:0340 c(3)l 2 cl + 0:577 cLL + 0:00878 L 0:0224 0:0372 c(3)l 2 c6 c6 2 cH 2 c6 2 = 0:0287 cW B + 0:170 cT 0:0741 L c(3)l 2 + 0:338 cLL (3)l 2 0:0282 L ce 0:0194 R2 + 0:000696 c3W 2 0 0 0 cW W 2 cW W 2 0 0 0 cW W 2 c(3)l 2 + 1:69 L ! e eh (250 GeV) 0 = 0:0139 cW W 2 0:0291 2 c(3)l (3)l 2 (3)l 2 (3)l 2 2 2 2 2 2 2 2 2 2 2 (2) Polarized ( 0:8; 0:3) 250 GeV (3) Polarized ( 0:8; 0:3) 500 GeV e+e ! Zhh (polarized beam (-0.8, 0.3) at 500 GeV) =0:912 + 0:173 cW B + 0:0339 cBB 2 0:312 cT 2 0:213 cH 2 (3)l cl + 0:417 cLL + 2:13 L 2 1:36 R c e 2 0:0345 c6 2 = 0:0420 cW B + 0:172 cT 0:0740 L c(3)l 2 + 0:343 cLL (3)l 2 0:0306 L 0:0354 cW B + 0:173 cT 0:0364 L c(3)l 2 + 0:346 cLL (3)l 2 (B.16) (B.17) (B.18) (B.19) (B.20) (B.21) (B.22) HJEP05(218) 3. Angular observables. Here we set the SM value of sin e Unpolarized beam at 240 GeV A 1 = + 0:00180 0:0000708 L 0:0000708 L 0:00000364 R A A A A Ac 1;c 2 = 0:00755 + 0:00953 (3) = 0:0136 + 0:0151 c(3)l c(3)l 2 2 cW W 2 cW W 2 2 2 0:0637 L cW W c(3)l 2 0:119 L cW W c(3)l 2 c(3)l 2 c(3)l 2 c(3)l 2 c(3)l 2 cl 2 0:0121 cW B 2 0:00285 cBB 2 (3)l cl ce + 0:0322 cLL + 0:0112 L2 + 0:0130 R 2 2 0:0212 cW B 2 0:00462 cBB 2 (3)l 2 0:0272 0:0518 2 cW B 2 cW B 2 (3)l cW B 2 (3)l e 2 e 2 e 2 2 2 0:000662 cBB 0:000803 cBB 2 2 0:000304 cBB 2 c e 2 (B.23) (B.24) (B.25) (B.26) (B.27) (B.28) (B.29) (B.30) (C.1) (4) = 0:0919 + 0:00444 + 0:000179 0:0000372 L 0:0000372 L 0:00000192 R C Normalized correlation matrices The normalized correlation matrix for the 6D operators is de ned as Here ci and cj run over all Wilson coe cients. Obviously the correlation matrix is sym 2 2 . cW B cl c L e R 0.179 0.114 0.938 cl 0.34 0.241 0.228 e 0.0494 c6 1 t at ILC500. g(c1; c2; ; cm) / exp i(cj ) = exp( 2 n X { 30 { t at ILC. D Parameter marginalization in 2 The introduction on parameter marginalization in 2 can be found in various lecture notes (see, e.g., [59]). Below we will simply introduce this method. Let's consider n independent observables O1; O2; ; On, with their measured values satisfying Gaussian distribution. These observables are all linearly dependent on m parameters or Wilson coe cients C T = fc1; c2; ; cmg, namely, Oi = Oi(c1; c2; ; cm) with i = 1; ; n and m n. Then their probability distribution function (PDF) is given by f ( 1 ; 2 ; n) = n exp n X Here i represents a normalized deviation from the measured value. f ( 1 ; 2 ; n) can be converted to the PDF of the m parameters 1 c3W 1 (D.1) (D.2) is a quadratic function of CT = fc1; c2; parameters which is symmetric. ; cmg. M is the correlation matrix of the m The marginalized distribution for a single parameter, say, cm, is de ned as Z gM (cm) = dc1dc2 dcm 1g(c1; c2; ; cm) Separating cm from the other m 1 paramters we have with CX = (c1 c2 cm 1)T . X, Z and y are the entries of the correlation matrix with dcm g(c1; c2; ; cm) = 1 2 = CT M C (D.3) (D.4) (D.5) (D.6) (D.7) (D.8) (D.9) (D.10) (D.11) 2 = yc2m + cmCXT Z + cmZT CX + CXT XCX M = " X Z # ZT y : 2 = yc2m + cmCX0T Z0 + cmZ0T CX0 + C0TX X0CX0 = yc2m + 2cm = m 1 X c0izi0 + m 1 X x0ic0i2 i=1 c2m + X x0i c0i + i=1 m 1 i=1 cmzi0 2 x0 2(cm) = y z0 1 . . . .. 7 . 77 : zm0 1 7 x0m 1 zm0 15 y The (m 1) (m 1) matrix X can be diagonalized by taking a unitary transformation CX ! CX0 . In the new parameter basis, we have Here we de ne CX0 = (c01 c0 2 (z10 z20 zm0 1)T . Integrating out CX0 , we obtain c0 m 1)T , X0 = x0m 1 ) and Z0 = It de nes the marginalized PDF of cm as (C0TX ; cm) as M 0 = " X0 Z0# Z0T y Taking a further step, let's de ne the correlation matrix in the new parameter basis The determinant of the correlation matrix M can be calculated as With this relation, we immediately obtain det M = det M 0 = Y x0i : i=1 2 = c2m ddeettMX in the m-dimensional space. case, 2 is de ned as This relation indicates that, given a con dence level for the 2 analysis, the constraints for cm is completely determined by the correlation matrix M . There is a geometric interpretation regrading this. Eq. (D.8) de nes a (m 1)-dimensional ellipsoid in a m-dimensional space which is expanded by CT = (CXT ; cm). Integrating out CX0 is equivalent to imposing the conditions c0i + cmzi0 = 0 or equivalently, the conditions @@c0i2 = 0, for i = 1; x0 i Therefore, the marginalization of CX is simply a projection of the ellipsoid to the cm axis ; m HJEP05(218) The discussions above can be generalized to the case with multiple variables. In this 2 = (CXT CYT ) " X Z ZT Y # " CX CY # with CX = c1; ; ck representing the k parameters to marginalize. Here X and Y are k k and (m k) matrices, respectively. With this setup, we have 2 = CXT XCX + CXT ZCY + CYT ZT CX + CYT Y CY CX is marginalized by integrating out the rst term, yielding 2 = CYT (Y ZT X 1Z)CY Eq. (D.16) describes the correlation among the parameters in CY . At a given C.L., the value of 2 depends on the number of parameters in CY . If CY contains one parameter only, say, cm, eq. (D.16) is reduced to eq. (D.9), with 2 = 1 at 1 C.L. Again marginalizing CX is equivalent to imposing the conditions @ 2=@ci to Eq. (D.15), with i running from 1 to k. It can be geometrically interpreted as a projection of a (m 1)-dimensional ellipsoid in a m-dimensional space to its (m k)-dimensional subspace, which are expanded by (CX ; CY ) and CY , respectively. The geometric interpretation of parameter marginalization in a simple model is presented in gure 10. In this example there are two free parameters, say, c1 and c2, with c2 being marginalized. Eq. (D.8) de nes an one-dimensional ellipsoid or ellipse at the c1 marginalization is simply a projection of the ellipse to the c1 axis. Here the size of the c2 ellipse is determined by the 2 value, which is equal to one at 1 C.L. In gure 10, the allowed range for c1 with a marginalized c2 is indicated by the brown line ending at the purple lines. As a comparison, if c2 is turned o , the constraint for c1 becomes stronger, which is denoted by the brown dashed line ending at the blue ellipse. (D.12) (D.13) (D.14) (D.15) (D.16) E 2D 2 analysis. 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Wen Han Chiu, Sze Ching Leung, Tao Liu, Kun-Feng Lyu, Lian-Tao Wang. Probing 6D operators at future e−e+ colliders, Journal of High Energy Physics, 2018, 81, DOI: 10.1007/JHEP05(2018)081