Measuring the trilinear Higgs boson self--coupling at the 100 TeV hadron collider via multivariate analysis
Jubin Park Jung Chang Kingman Cheung Jae Sik Lee
hep-ph




IUEP-HEP-20-01

 




#1#1/





^1 IUEP, Chonnam National University, Gwangju 61186, Korea 

^2 Department of Physics, Chonnam National University,
Gwangju 61186, Korea

^3 Physics Division, National Center for Theoretical Sciences,
Hsinchu, Taiwan 

^4 Division of Quantum Phases and Devices, School of Physics, 
Konkuk University, Seoul 143-701, Republic of Korea 

^5 Department of Physics, National Tsing Hua University,
Hsinchu 300, Taiwan 









We perform a multivariate analysis of Higgs-pair production 
via the decay channel  HH → bb̅γγ at the
future 100 TeV pp collider to determine
the trilinear Higgs self–coupling (THSC) λ_3H, which takes
the value of 1 in the standard model.

We consider all known background processes.
For the signal we adopt the most recent event generator of
POWHEG-BOX-V2 to exploit the NLO distributions 
for Toolkit for Multivariate Data Analysis (TMVA).


Through the technique of Boosted Decision Tree (BDT) analysis trained
for λ_3H=1,
compared to the the conventional cut-and-count approach,
the signal-to-background ratio improves tremendously 
from about 1/10 to 1
and the significance can reach up to 20.5 with a luminosity of 3 ab^-1
without including systematic uncertainties.

In addition, by implementing a likelihood fitting of the signal-plus-background
M_γγ b b distribution with optimized bin sizes, 
it is possible to determine
the THSC with the precision of 7.5% at 68% CL
even at the early stage of 100 TeV hadron collider with 3 ab^-1.




Measuring the trilinear Higgs boson self–coupling 
at the 100 TeV hadron collider via multivariate analysis

    
Jubin Park^1,2,
Jung Chang^2, Kingman Cheung^3,4,5, 
and Jae Sik Lee^2,1
    September 11, 2020
=============================================================================================================





§ INTRODUCTION
 

Since the discovery of the 125 GeV Higgs boson in 2012 at the
LHC <cit.>,
we have been looking for a clear signal or 
even a hint of new physics beyond the Standard Model (SM) but without much success.

Moreover, after completing the Runs I and II at the LHC, 
it turns out that the 125 GeV Higgs boson is best described as 
the SM Higgs boson <cit.>,
although there is an upward trend in the overall signal
strength <cit.>.

Under this situation,  one of the most solid avenues to explore for new physics
is to measure the Higgs potential which could be significantly different from
that of the SM.

Higgs-boson pair production at the high-luminosity and/or high-energy
hadron colliders provides a very useful way to probe the Higgs potential
via the investigation of the trilinear Higgs self-coupling 
(THSC) <cit.>.

The specific decay modes considered are: bb̅ bb̅ <cit.>,
bb̅γγ <cit.>,
bb̅τ^+τ^- <cit.>,
bb̅ W^+W^- <cit.>, and some combinations
of these channels <cit.>.

Higgs-boson pair production also has been vastly studied in 
models beyond the SM <cit.>.

The current limits on the THSC in units of λ_3H, which takes
the value of 1 in the SM, are

-5.0 < λ_3H < 12  from ATLAS <cit.>
and -11.8 < λ_3H < 18.8  from CMS <cit.>
at 95% confidence level (CL). 

At the high-luminosity option of the LHC running at 14 TeV (HL-LHC)
with an integrated luminosity of 3 ab^-1, 
a combined ATLAS and CMS projection of the 68% CL interval 
is 0.57 < λ_3H < 1.5 
without including systematic uncertainties <cit.>.

On the other hand, at the International Linear Collider (ILC) operated at
1 TeV can reach the precision of 10% at 68%  CL with 
an integrated luminosity of 8 ab^-1 <cit.>.



In this work, we perform a multivariate analysis of Higgs-pair production 
in HH → bb̅γγ channel at the 
100 TeV hadron collider.

In our previous work,   
based on the conventional cut-and-count analysis,
it was shown that the THSC can be measured with about 20% accuracy
at the SM value with a luminosity of 3 ab^-1 <cit.>.




In this Letter, with the use of the BDT method 
closely following Ref. <cit.>,
we show that the THSC can be measured with 
a precision of 7.5% at 68% CL at the 100 TeV hadron collider
assuming 3 ab^-1 luminosity,
which is superior to the accuracy expected
at the 1 TeV ILC even with 8 ab^-1.



§ EVENT GENERATION AND TMVA ANALYSIS


The Higgs bosons in the signal event samples
are generated on-shell with a zero width by
 <cit.> with
the damping factor 𝚑𝚍𝚊𝚖𝚙 set to the default value of 250
to limit the amount of hard radiation. 

This code provides NLO distributions matched to a parton shower 
taking account of the full top-quark mass dependence. 

The signal cross section at NNLO order in QCD is calculated according to
σ^ NNLO ( λ_3H )=
K^ NNLO / NLO_ SM σ^ NLO  (λ_3H)
using σ^ NLO(λ_3H) from POWHEG-BOX-V2
and  K^ NNLO/NLO_ SM=1.067 <cit.>
[
According to the recent N3LO calculations <cit.>, 
the signal cross section is further enhanced by the amount of 2.7% which 
would hardly affect our conclusion, or rather strengthen our results.]
in the FT
approximation in which the full top-quark mass dependence is considered only
in the real radiation while the Born improved Higgs Effective Field Theory
is taken in the virtual part.

And then, 
the  code <cit.> is used for the decay of
both Higgs bosons into two bottom quarks and two photons.




For generation and simulation of backgrounds,
we closely follow Ref. <cit.>
[Specifically, the multi-variate MV1 b-tagging algorithm 
with ϵ_b=0.75 is taken together with P_c→ b=0.1, 
P_j→ b=0.01, and P_j→γ=1.35× 10^-3 <cit.>.], 
except for the use
of the post-LHC PDF set of CT14LO <cit.> 
for non-resonant backgrounds.
Furthermore,
for the two main non-resonant backgrounds of bb̅γγ and cc̅γγ,
we use the merged cross sections and distributions
by MLM matching <cit.>
with xqcut and Q_ cut set to 20 GeV and 30 GeV, respectively.

For the remaining non-resonant backgrounds,
we are using the cross sections and distributions obtained by applying
the generator-level cuts as adopted in Ref. <cit.>
which might provide more reliable and conservative estimation of the non-resonant
backgrounds
containing light jets <cit.>.

For parton showering and hadronization,
 <cit.> is used both for
signal and backgrounds.

Finally, fast-detector simulation and analysis are
performed using  <cit.>
with the  template.

All the signal and backgrounds are summarized in Table <ref>,
together with information of the corresponding event
generator, the cross section times the branching ratio and 
the order in QCD, and the Parton Distribution Function (PDF) used.











A multivariate analysis is performed using TMVA <cit.>
with ROOTv6.18 <cit.>. 

After applying a sequence of event selections
as in Table <ref>,
we choose the following 8 kinematic variables for TMVA:

    M_bb ,   P_T^bb ,   Δ R_bb ;  M_γγ ,   P_T^γγ ,   Δ R_γγ ;  M_γγ bb ,   Δ R_γ b .


The judicious choice of the two photons or two b quarks for the above TMVA variables
has been made as in  <cit.>.

We also refer to Ref. <cit.> for the details of our TMVA setup and
analysis. And we choose  BDT for our analysis since
the BDT-related methods show higher performance
with better signal efficiency and stronger background rejection.









§ RESULTS


In the left panel of Fig. <ref>, we show the BDT responses obtained using
BDT trained for λ_3H=1 which is dubbed as BDT_ SM.

By validating the BDT distributions for the training sample (dots with error bars)
with those for the test sample (histogram), we check that BDT_ SM is not 
overtrained. 

In the right panel of Fig. <ref>, using BDT_ SM,
we show the behavior of signal and background efficiencies (inset)
and significance 
Z = √( 2 ·[ ( (s+b) ·ln( 1+ s/b) - s )   ] )
with s and b being the numbers of signal and background events
as functions of the cut value on BDT response.

The significance can reach up to 20.50 when the BDT response is cut
at 0.216, at which, the signal and background efficiencies are
0.48 and 1.58× 10^-4, respectively.

We denote by vertical lines the positions of the optimal cut on 
the BDT response which maximizes the significance.







In Table <ref>, we present the
expected number of signal and background events at the 100 TeV hadron 
collider assuming 3 ab^-1 using BDT_ SM with the BDT response cut of 0.216.

We show the four representative values of λ_3H for signal
and the backgrounds are separated into three categories.
For comparisons, we also show the results obtained using the cut-and-count
analysis <cit.>.

In the last column, we additionally present the effective luminosity (Eff. Lumi.) for each
of signal and background samples.

In the tt̅ and tt̅γ backgrounds,
the first (second) number is the effective luminosity
when the two top quarks decay fully (semi-) leptonically.




We find about 550 signal and 550
background events for λ_3H=1. Comparing to the results using
the cut-and-count analysis <cit.>,
the number of signal events
decreases by only 19% while the number of backgrounds by almost 90%, resulting in 
an increase in significance from 8.44 to 20.50.

Note that the composition of backgrounds changes drastically by the use of BDT.
In the cut-and-count analysis, the non-resonant background is about two times 
larger than the single-Higgs associated background. 
While, in the BDT analysis,
the single-Higgs associated background is more than four times  
larger than the non-resonant one
and tt̅ associated background becomes negligible.

Note that we generate relatively smaller number of events for
the cc̅γγ, cc̅jγ, and
bb̅jj backgrounds since we observe that they quickly decrease when
the BDT response cut approaches to the point Z_ max of 0.216
[In fact,
there are some differences in kinematic distributions among 
the non-resonant backgrounds. For example, 
the cc̅γγ background is more populated in the
region of Δ R_bb>3 compared to the bb̅γγ one.].

Specifically, the bb̅ j j background vanishes
for the BDT response cut larger than 0.2.

Otherwise, we generate enough number of events considering the assumed luminosity of
3 ab^-1.




First, we try to determine the THSC considering the total number of events.
As shown in the left panel of Fig. <ref>,
we find that the THSC can be measured with about 11% accuracy
at the SM value which is about two times better than the result
based on the conventional cut-and-count analysis <cit.>.
However, there is a second solution around λ_3H=6.5.

To lift up the two-fold ambiguity,
we implement a likelihood fitting of the signal-plus-background
M_γγ bb distribution
and find the second solution is ruled out 
 by more than 8σ confidence,
see the right panel of Fig. <ref>.






  To improve the sensitivity of the THSC around the SM value 
  and to tame the statistical fluctuation due to the limited
  size of the MC samples, we repeat the likelihood fitting of
  M_γγ bb distribution by optimizing the bin size
  between 1/20GeV and 1/60GeV.
Finally, we find that
the THSC can be determined with a precision of 7.5% at 68% CL
as shown in the left panel of Fig. <ref>. 

In the right panel of Fig. <ref>, M_γγ bb distributions
are shown for
the THSC at the SM value and for the two values
deviated by 1σ.





By now, we have considered only the statistical uncertainties which
may eventually dominate the total uncertainties.
Before concluding, we would like to discuss the effects of 
systematic uncertainties which could be important
at the early stage of 100 TeV hadron collider.
The systematic uncertainties might be taken into account by 
considering the variance of background σ_b^2 <cit.>.
In this case, the error due to systematic uncertainties is proportional to the number
of background or σ_b ∝ b.












We find that the THSC precision of 7.5%-18%  at 68% CL
while varying σ_b/b between 0
and 0.2, see Fig. <ref>
[Incidentally, by measuring only the total number of events,
the precision becomes worse to 11%-30%.].


Finally, before we end this section, in Table <ref>,
we show the relative importance of the variables

that we employed in this BDT analysis.
We observe that
the two most important variables are Δ R_bb and Δ R_γγ,
which is consistent with our previous cut-and-count analysis
<cit.>.







§ CONCLUSIONS:


Higgs-pair production is one of the most useful avenue to probe the EWSB
sector. We have studied in great details, with the help of
machine learning, the sensitivity of measuring the THSC λ_3H
that one can expect at the 100 TeV pp collider with an integrated
luminosity 3 ab^-1. With TMVA one can improve
the signal-to-background ratio for λ_3H=1
to 1:1 compared with the ratio 1:10 obtained in the conventional
cut-and-count approach.
Furthermore, the significance of such a signal jumps to 20.

Other than determining the THSC by measuring the total number of events,
one can also improve the sensitivity and lift the two-fold degeneracy
by implementing a likelihood fitting of the signal-plus-background
M_γγ b b distribution with optimized bin sizes. The THSC
can be determined with a precision of 7.5% at 68% CL
with 3 ab^-1, which is indeed
better than the ILC running at 1 TeV with 8 ab^-1.

Extrapolating our result
conservatively, we expect that one can achieve the precision better than
∼ 2% with 30 ab^-1.



Note added: After the completion of our work, we learned 
a similar analysis performed considering various systematic 
uncertainties rigorously <cit.>.
They found the combined
precision of 2.9%-5.5% with 30 ab^-1 at 68% CL
which is in a good quantitative agreement with our results.




§ APPENDIX


section.equation






§ MORE ON THE CC̅ΓΓ, CC̅JΓ, AND
BB̅JJ BACKGROUNDS


For this work,
we generate relatively smaller number of events for
the cc̅γγ, cc̅jγ, and
bb̅jj backgrounds which may lead to underestimation
of the relevant backgrounds.

The cc̅γγ and cc̅jγ backgrounds might be
negligible since, taking account of the fake rates P_c→ b and
P_j→γ, the cross sections are smaller than
that of the bb̅γγ background by about an order of magnitude.

On the other hand, our estimation of the bb̅jj background
could be unreliable due to the limited size of the MC sample.
Here we try to estimate the background yield based on the current sample.




Precisely, we study the behavior of the bb̅ jj background yield Y_bb̅jj versus
the BDT response cut.

First we observe that, based on the current bb̅ jj MC sample,
our estimation of the background results in 0 when BDT Cut >0.19, see
the upper-left panel of Fig. <ref>.

To extrapolate to the region with BDT Cut >0.19, we implement
a linear fitting to log Y_bb̅jj, see the solid line
in the upper-right panel of Fig. <ref>.

And we find that

    Y_bb̅jj=13.7^+2.2 (5.2)_-1.9 (3.7)

at 68(95)% CL as shown in the lower panel of Fig. <ref>.

Taking the 1σ upper value of 15.9,
the number of total background increases by
the amount of about 3% which hardly affect
our main results significantly.

Incidentally, we note that  the
jjγγ background survives though its cross section
is smaller than that of the bb̅γγ one
by about two orders of magnitude taking account of
the fake rate P_j→ b. This is because
its kinematical distributions quite resemble to those of the signal.
For example, compared to other non-resonant backgrounds,
we find that it is quite populated
in the region of Δ R_bb 2 where
most signal events are located.





§ SUPPLEMENTAL MATERIALS


In this appendix, we present the
normalized distributions of the eight kinematic variables
for the SM signal with λ_3H=1 (black solid) and 
the six non-resonant backgrounds 
after applying the event preselection cuts 1-5 in Table <ref>,
see Fig. <ref>.

For M_bb ,γγ and P_T^bb ,γγ, in terms of P_T,
we choose the least energetic two photons or two b quarks
while the most energetic ones are chosen for Δ R_bb,γγ and
M_γγ bb. For Δ R_γ b, on the other hand, we choose the least
energetic b and the next-to-the-least energetic photon.


















§ ACKNOWLEDGMENT

This work was supported by the National Research Foundation of Korea 
Grant No. NRF-2016R1E1A1A01943297 (J.C., J.S.L., J.P.), 
No. NRF-2018R1D1A1B07051126 (J. P.), 
and by the MoST of Taiwan under Grant No. 107-2112-M-007-029-MY3 (K. C.).



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ATLAS collaboration,
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  A. J. Barr, M. J. Dolan, C. Englert, D. E. Ferreira de Lima and M. Spannowsky,
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  S. Homiller and P. Meade,
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bbaa_atlas17
ATLAS Collaboration,
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bbtata



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bbWW



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  V. Martín Lozano, J. M. Moreno and C. B. Park,
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decaycombined



  M. J. Dolan, C. Englert and M. Spannowsky,
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  F. Goertz, A. Papaefstathiou, L. L. Yang and J. Zurita,
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  M. J. Dolan, C. Englert, N. Greiner, K. Nordstrom and M. Spannowsky,
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A. Adhikary, S. Banerjee, R. K. Barman, B. Bhattacherjee and S. Niyogi,
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S. Banerjee, C. Englert, M. L. Mangano, M. Selvaggi and M. Spannowsky,
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  J. H. Kim, M. Kim, K. Kong, K. T. Matchev and M. Park,
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comb_atlas18
ATLAS Collaboration,
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ATL-PHYS-PUB-2018-053.

dihiggs_np



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  J. M. No and M. Ramsey-Musolf,
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  C. R. Chen and I. Low,
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  A. Papaefstathiou,
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Aad:2019uzh 
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Sirunyan:2018two 
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Cepeda:2019klc 
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Fujii:2015jha 
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Braathen:2019zoh 
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Chang:2018uwu 
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Chang:2019ncg 
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Heinrich:2017kxx
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Heinrich:2019bkc
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n3lo


L. B. Chen, H. T. Li, H. S. Shao and J. Wang,
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Artoisenet:2012st
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Contino:2016spe
R. Contino, D. Curtin, A. Katz, M. Mangano, G. Panico, M. Ramsey-Musolf, G. Zanderighi
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Mangano:2006rw
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Alwall:2014hca
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TMVA2007
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ROOT
Rene Brun and Fons Rademakers, ROOT - An Object Oriented Data Analysis Framework,
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Cowan:2010js
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Mangano:2020sao
M. L. Mangano, G. Ortona and M. Selvaggi,
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