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
KunFeng Lyu 0 2 5
LianTao 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 FCCee 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 sixdimensional (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 CPeven and 17 are CPodd. 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
yettobeexplicitlyestablished 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 FCCee analysis. Refs. [25, 26] took similar strategies, with the results presented in
the \ "scheme and in the 6D operatorscheme, 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 FCCee. 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 CPeven
and 3 CPodd ones. In this article, we focus only on the CPeven 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 wavefunction 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 wellselected 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. Zassociated dihiggs process.
As the beam energy increases, diHiggs 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 CPeven,
! 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
nondegenerate 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. Diboson process. The diboson 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 FCCee
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 statisticalerrorlike. 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 wellprobed 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 \semiconservative" (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 10dimensional
ellipsoid in an 11dimensional 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 \semiconservative" 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 FCCee 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 diphoton 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 diphoton 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, FCCee, 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
GigaZ 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, FCCee 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 FCCee are circular e e+ colliders with nonpolarized beams. Bene tting from
a larger integrated luminosity at Z pole, the sensitivities at FCCee 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 FCCee, because of its relatively small luminosity at 250 GeV and the
lack of data at Zpole. 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 FCCee ones in the
optimistic case. In the conservative case, the ILC500 performance becomes comparable to or
even better than the CEPC and FCCee ones. This results in a smaller di erence between
the two kinds of sensitivity interpretations at ILC, compared with the ones at CEPC and
FCCee, as is indicated in gure 7. On the other hand, the data at GigaZ 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 FCCee
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 fulltime 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 FCCee, 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 Zpole 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 FCCee. 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 FCCee. This is
due to the e e+
! Zhh production, an observable which is not available at CEPC and
FCCee. 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 FCCee , their weakness in probing O6 could be mitigated somewhat
by combining with the LHC data for diHiggs 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 nonlinear c6 terms, e.g., the oneloop
quadratic term induced by the Higgs selfenergy 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
looplevel 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
modeldependent. 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 HLLHC. 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 spacetime. This spacetime, 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 looplevel
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 leftupper 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 (leftupper 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, FCCee 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 treelevel
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 FCCee 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 FCCee 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 Zpole, 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 FCCee ones. Moreover, compared
to CEPC and FCCee, 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
FCCee [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, FCCee 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. DESC0013642.
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 mdimensional 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 mdimensional
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 mdimensional 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 onedimensional 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. The coordinate axes are in the unit of (TeV) 2.
2 analysis. The coordinate axes are in the unit of (TeV) 2.
2 analysis. The coordinate axes are in the unit of (TeV) 2.
FCCee and ILC. The coordinate axes are in the unit of (TeV) 2. In the \optimistic" analysis, only
two 6D operators are turned on. In the \conservative" analysis, all 6D operators listed in table 1
are turned on, whereas the irrelevant Wilson coe cients are marginalized.
Open Access.
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