This page describes the parametrization adopted in CMS for Higgs theoretical lineshape at high mass. This was used in CMS-HIG-13-031 for the H -> ZZ -> 2l2l' shape parametrization, and can be used for H -> ZZ -> 2l2q case.

It was obtained using, for both gluon-gluon fusion and VBF, by running the relevant generators and extracting the interference shape, which was then fitted with an ad-hoc function.

The parametrization depends on the mass of the Higgs boson, on the width and on its cross section.

Many more details can be found in this thesis.

Obtaining the interference lineshape.

For the gluon-gluon case, we obtained the lineshape using GG2VV version 3.1, while for VBF we used PHANTOM version 1.2.3. The procedure for extracting the interference follows the recommendation of the HXSWG documented in Yellow Report 3, chapter 12.1.2 page 194. We use the generators to obtain the signal, the background and the |signal + background|^2 distributions, and then subtract from the latter the two formers. To move from the LO generator shape to the NNLO distribution, we use the intermediate approach documented in the YR3, and then evaluate the systematic effect of this choice by using the additive and multiplicative approaches.

We produced several samples for mH > 400 GeV, each one with a statistics corresponding to approximately 1M fb-1 for a SM Higgs boson. All samples have been generated using the CTEQ6L1 LO PDF, and with running renormalization and factorization scales of mH/2. The following cuts have been applied at generator level:

  • pT > 5 GeV and |eta| < 2.7 for electrons and muons
  • m_4l > 100 GeV
  • p_T,ll > 1 GeV which is included in the generator settings for computational stability and has a negligible effect for the range of Higgs masses under study
For the VBF samples generated with PHANTOM, the following additional cut was used:

* p_T,jet > 10 GeV, |eta_jet| < 6.5, m_jj > 30 GeV

In PHANTOM, processes corresponding to signal only or background only amplitudes cannot be calculated, because they spoil the regularization of the continuum processes at high masses. In order to access the mass distribution of each single contribution, the following relation is used:

   \begin{equation}<br />P_{tot} = \mu \times P_S + \sqrt{\mu} \times P_I + P_B<br />   \end{equation}

One can construct a three-equation linear system (for every value of the Higgs mass and width), each equation corresponding to the relation 1 evaluated at a certain signal strength. The signal, background and interference probabilities (P_S , P_B and P_I ) can be then extracted by inverting the system.

We performed the generation in the following grid of mH and C' (the scale factor to normalization and width):

  • mH: 400,500,600,700,800,900,1000
  • C'^2: 0.2,0.4,0.6,0.8,1.0 (gg2VV) e 0.1,0.3,0.5,0.7,1.0 (Phantom)

Here are some typical examples of signal, background and total lineshapes obtained with GG2VV for several values of mH, and for two values of the cross section (C' scales cross section and width with respect to the SM):


And here are some typical examples of signal, background and total lineshapes obtained with PHANTOM for several values of mH, and for two values of the cross section (C' scales cross section and width with respect to the SM):


Interference parametrization

From the plots above, one can see that the interference is constructive below the pole mass, and destructive above, albeit not completely symmetric. For this reason, it's effect on the total cross section is negligible in this specific channel. The effect becomes more important at higher mass, and is actually neglected for mH < 400 GeV. An analytical parametrization is used to describe the shape of the interference, which is built from simple assumptions. If the signal part in the amplitude is described with a real pole and the background is modeled with a simple exponential, which is a good approximation in the region mZZ > 300 GeV, we can write:

   \begin{equation}<br />|S + B|^2 = |C_S \frac{1}{m^{2}_{ZZ}-m^{2}_{H}} + C_{B} e^{-\alpha m_{ZZ}} |^2 = |S|^2 + |B|^2 + I<br />   \end{equation}

One can then move to the complex pole scheme for signal with the substitution m_H^2 -> m_H^2 -im_HGamma, which leads to the following expression for the interference

   \begin{equation}<br />I(m_{ZZ},m_H,\Gamma,\alpha) = Re(\frac{e^{-\alpha m_{ZZ}}}{m^{2}_{ZZ}-m^{2}_{H} + im_H \Gamma })<br />   \end{equation}

Here we show some typical fits to the interference lineshapes


The plots show a large uncertainty in the lower mH part of the distribution. This is due to the fact that in that region we need to subtract two very large distributions (S+B and B) from each other in order to get a very small number. In order to get the total S+I parametrization, we fit the distribution with the following function:

   \begin{equation}<br />S_{corr} (m_{ZZ},m_H,\Gamma,\alpha,r) = S(m_{ZZ},m_{H},\Gamma) - r \cdot I(m_{ZZ},m_H,\Gamma,\alpha)   <br />   \end{equation}

The parameter r is fitted on the S+I distributions, leaving all the other parameters fixed.

The final S+I lineshape parametrization, compared with the generator level S+I lineshape, is shown here:


The parameters of the analytical description are then interpolated between the different mH and C' samples to get a continuous description.

RooFit code

A RooFit class is provided describing the heavy Higgs lineshape in presence of signal-background interference. Note that no heavy Higgs - h(125) interference is taken into account.

The source code, the instructions to compile it and an example of usage are included in RooSigPlusInt.tar.gz.

-- MarioPelliccioni - 2015-05-15

Topic attachments
I Attachment History Action Size Date Who Comment
PNGpng Interf_fit.png r1 manage 131.8 K 2015-05-15 - 11:48 MarioPelliccioni Fit to the interference lineshape
PNGpng PHANTOMshapes.png r1 manage 84.4 K 2015-05-15 - 11:27 MarioPelliccioni Higgs lineshapes obtained from PHANTOM
Unknown file formatgz RooSigPlusInt.tar.gz r1 manage 20.7 K 2015-05-29 - 10:47 StefanoCasasso Source code of the class implementing the heavy Higgs lineshape
PNGpng S_I_lineshape.png r1 manage 140.3 K 2015-05-15 - 11:57 MarioPelliccioni Higgs theoretical lineshape (generator and parametrization)
PNGpng gg2VVshapes.png r1 manage 89.6 K 2015-05-15 - 11:21 MarioPelliccioni Higgs lineshapes obtained from gg2VV
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Topic revision: r5 - 2018-09-13 - WeiminSong
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