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HeavyN
: The Standard Model + Heavy Neutrinos at NLO in QCD
Contact Author
Richard Ruiz
- Institute of Nuclear Physics Polish Academy of Science (IFJ PAN)
- richard.physics AT gmail.com
In collaboration with:
- Alva and T. Han [ 1 ]; C. Degrande, O. Mattelear, and J. Turner [ 2 ]; S. Pascoli and C. Weiland [ 3, 4 ]; and V. Cirigliano, W. Dekens, J. de Vries, K. Fuyuto, E. Mereghetti [ 5 ].
Usage resources
- For detailed instructions and examples on using the HeavyN UFO libraries, see C. Degrande, et al, arXiv:1602.06957 and S. Pascoli, et al, arXiv:1812.08750 .
- For heavy neutrinos in vSMEFT, see V. Cirigliano, et al, arXiv:2105.11462.
- See Validation section below for additional information
Citation requests
- For studies of heavy Majorana neutrinos, please consider citing [ 6 ] for the Lagrangian and [ 1, 2 ] for the Majorana FR/UFO files.
- For studies of heavy Dirac neutrinos, please also consider citing [ 4 ].
- For studies of heavy neutrinos in vSMEFT, please consider citing [ 5 ].
Model Description
Majorana N
This effective/simplified model extends the Standard Model (SM) field content by introducing three right-handed (RH) neutrinos, which are singlets under the SM gauge symmetry (no color, weak isospin, or weak hypercharge charges). Each RH neutrino possesses one RH Majorana mass. After electroweak symmetry breaking, the Lagrangian with three heavy Majorana neutrinos Ni (for i=1,2,3) is given by [ 6 ]
\begin{equation} \mathcal{L} = \mathcal{L}_{\rm SM} + \mathcal{L}_{N} + \mathcal{L}_{N~\text{Int.}} \end{equation}
The first term is the Standard Model Lagrangian. In the mass basis, i.e., after mixing with active neutrinos, the heavy Majorana neutrinos' kinetic and mass terms are
\begin{equation} \mathcal{L}_{N} = \frac{1}{2}\overline{N_k} i\!\not\!\partial N_k - \frac{1}{2}m_{N_k} \overline{N_k}N_k, \quad k=1,\dots,3, \end{equation}
and its interactions with the Weak gauge and Higgs bosons are given by
\begin{eqnarray} \mathcal{L}_{N~\text{Int}} = &-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}\gamma^{\mu}P_{L}\ell^{-} +{\rm H.c.} \\ &-&\frac{g}{2\cos\theta_W}Z_{\mu}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}\gamma^{\mu}P_{L}\nu_\ell +{\rm H.c.} \\ &-&\frac{g m_N}{2 M_W} h \sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}P_{L}\nu_\ell +{\rm H.c.} \end{eqnarray}
Neutrino masses (mNk) and mixing parameters (Vlk) between heavy mass eigenstate and (active) flavor eigenstates are taken to be independent, phenomenological parameters. This allows for maximum flexibility and model independence when calculating rates. Therefore, some care is required by the user. The lepton number- and flavor-violating interactions of the Lagrangian allow for modeling of the Type I, Inverse, and Linear seesaw mechanisms at both lepton, hadron, and lepto-hadron colliders.
Dirac N
The Dirac variant of the HeavyN model file contains the same interaction Lagrangian as the Majorana case. The heavy Dirac neutrinos' kinetic and mass terms are [ 4 ]
\begin{equation} \mathcal{L}_{N} = \overline{N_k} i\!\not\!\partial N_k - m_{N_k} \overline{N_k}N_k, \quad k=1,\dots,3. \end{equation}
*New* vSMEFT at dimension six
In the HeavyN_vSMEFTdim6
UFO [ 5 ], the HeavyN
(Majorana) Lagrangian is extended by the dimension-six EFT operator
\begin{equation} \mathcal{L}_{H\nu e}^{(6)} = \frac{1}{\Lambda^2} \left(i \tilde{\varphi}^{\dagger} D_{\mu} \varphi\right) \, \bar \nu_R \gamma^\mu \,C_{H\nu l} ~ l_R + \mathrm{H.c.}\, , \end{equation}
where D_\mu
is the usual SM covariant derivative. \Lambda
is the EFT cutoff in GeV scale, C_{H\nu l}
is the dimensionless Wilson coefficient, and $l=e,\mu,\tau$
. Note that in Ref. [ 5 ], C_{H\nu l}
has dimension GeV^-2
. In the unitary gauge after and EW symmetry breaking, the leading contribution of this operator can be written in the neutrino mass basis as
\begin{equation} \mathcal{L}_{H\nu e}^{(6)} \approx \frac{g v^2}{2\sqrt{2}\Lambda^2} \sum_{\ell=e}^{\tau} \sum_{k=1}^{3} \,\overline{N_k} \left[\bar{C}_{H\nu l}\right]_{k \ell} \gamma^\mu P_R \ell_{R}\,W_\mu^+ \left(1+\frac{h}{v}\right)^2 +\text{H.c.} \end{equation}
Here k=1,\dots,3
runs over all three heavy neutrino mass eigenstates and $\ell=e,\mu,\tau$
runs over all lepton flavors. For the precise definition of the effective Wilson coefficient $\bar{C}_{H\nu l}$
, see Ref. [ 5 ].
QCD Corrections
The above Lagrangian with Goldstone boson couplings and in the Feynman Gauge was implemented into FeynRules 2.3.10. QCD renormalization and R2 rational counter terms were determined using NLOCT 1.02 and FeynArts 3.8. Feynman rules were collected into a single UFO, available below. In the UFO file, five massless quarks are assumed as are zero off-diagonal CKM matrix entries. For additional details, see [ 2 ] and references therein. These additions permit tree-level calculations at LO and NLO in QCD and loop-induced calculations at LO in QCD using MadGraph_aMC@NLO.
Model Files
Note: The only difference between NLO and LO libraries is the presence of additional (effective) O(a_s) Feynman rules. By definition the NLO libraries can compute tree-level processes at LO precision.
Majorana
- SM_HeavyN_NLO_UFO.tgz: Standalone UFO folder. Assumes massless bottom quark and tau lepton, diagonal CKM.
- SM_HeavyN_Gen3Mass_NLO.tgz: Standalone UFO folder. Assumes massive bottom quark and tau lepton, diagonal CKM.
- SM_HeavyN_CKM_AllMasses_LO.tgz: Standalone UFO folder. Assumes LO in QCD, with CKM elements (in radians), and all particle masses. Majorana equivalent of SM_HeavyN_Dirac_CKM_Masses_LO_UFO.
- heavyN.fr: Main model file. Relies on sm.fr (default FR model file) being declared elsewhere.
- heavyN_NLO.nb: Mathematica notebook file that generates UFO file from FeynRules model files. Allows user to also run quick sanity checks (optional) on model.
- SM_HeavyN_Files.tgz: Standalone package containing heavyN.fr, heavyN_NLO.nb, massless.rst (default FR file), diagonalCKM.rst (default FR file), and sm.fr (default FR file).
- SM_HeavyN_FilesWithUFO.tgz: Combination of SM_HeavyN_NLO_UFO.tgz and SM_HeavyN_Files.tgz.
Dirac
- SM_HeavyN_Dirac_NLO_UFO.tgz: Standalone NLO UFO libraries
- SM_HeavyN_Dirac_LO_UFO.tgz: Standalone LO UFO libraries.
- SM_HeavyN_Dirac_CKM_Masses_LO_UFO.tgz: Standalone LO UFO libraries, assuming non-diagonal CKM matrix, six quark masses, and three charged lepton masses
- heavyN_Dirac.fr: Main model file. Relies on sm.fr (default FR model file) being declared elsewhere.
*New* vSMEFT
- SM_HeavyN_vSMEFTdim6_NLO.tgz: Standalone NLO UFO folder. Assumes massless bottom quark and tau lepton, diagonal CKM.
- SM_HeavyN_vSMEFTdim6_XLO.tgz: Standalone LO UFO folder. Assumes massless bottom quark and tau lepton, diagonal CKM.
- heavyN_vSMEFTdim6.fr: Main model file. Relies on sm.fr (default FR model file) being declared elsewhere.
Download and Unpack
- To download any of the packages and unpack via the terminal, use the commands:
Majorana NLO
~/Path $ wget https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_NLO_UFO.tgz
~/Path $ tar -zxvf SM_HeavyN_NLO_UFO.tgz
Majorana LO
~/Path $ wget https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_FilesWithUFO.tgz
~/Path $ tar -zxvf SM_HeavyN_FilesWithUFO.tgz
Dirac NLO
~/Path $ wget https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_Dirac_NLO_UFO.tgz
~/Path $ tar -zxvf SM_HeavyN_Dirac_NLO_UFO.tgz
Dirac LO
~/Path $ wget https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_Dirac_LO_UFO.tgz
~/Path $ tar -zxvf SM_HeavyN_Dirac_LO_UFO.tgz
vSMEFT NLO
~/Path $ wget https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_vSMEFTdim6_NLO.tgz
~/Path $ tar -zxvf SM_HeavyN_vSMEFTdim6_NLO.tgz
vSMEFT LO
~/Path $ wget https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_vSMEFTdim6_XLO.tgz
~/Path $ tar -zxvf SM_HeavyN_vSMEFTdim6_XLO.tgz
Notes
- For instructions on using the HeavyN UFO, see C. Degrande, et al, arXiv:1602.06957 and S. Pascoli, et al, arXiv:1812.08750
- The flagship
HeavyN
UFO model contains 15 free parameters:- Three masses: mN1, mN2, mN3. Defaults are 300 GeV, 500 GeV, and 1 TeV, respectively.
- Three widths: WN1, WN2, WN3. Defaults are 0.303 GeV, 1.50 GeV, and 12.3 GeV, respectively.
- Nine real (no CP violation) mixing parameters: Vlk for l = e, mu, tau and k = 1,2,3. Default values are Vlk = Identity(3x3), i.e.,
Ve1 = Vmu2 = Vta3 = 1
and all others zero. - Note: VlN are restricted to be real in the model file.
- Note: Default parameters are set so "out-of-the-box" checks can be made with [ 1 ] and [ 2 ].
- For the Majorana file, particle identification (PID) codes for N1,...,N3, follow standard HEP MCPID codes: 9900012, 9900014, 9900016
- For the Dirac file, to avoid conflict with Pythia8, where the above PIDs are reserved for Majorana fields, the nonstandard HEP MCPID codes for N1,...,N3 are:9990012, 9990014, 9990016
- For the vSMEFT file, 10 additional model parameters are introduced:
- One EFT cutoff scale
Lambda
in units of GeV. - Nine Wilson coefficients coupling
N_k
tol
:CeN1, CeN2, CeN3, CmuN1, CmuN2, CmuN3, CtaN1, CtaN2, CtaN3
- Note: Default parameters are set such that
Lambda=1000
(GeV),CeN1=CmuN2=CtaN3=1
, and all other coefficients are zero
- One EFT cutoff scale
Validation
- The model file was validated at LO in [ 1 ]; see Tables 2 and 5 for further validation checks.
- The model file was validated at NLO in [ 2 ]; see Table 1 for further validation checks.
- For very large masses, this model has been constructed to satisfy the Goldstone Equivalence Theorem:
\begin{eqnarray} \text{BR}(N_i \rightarrow W^+ \ell^-) &=& \text{BR}(N_i \rightarrow W^- \ell^+) = \\ \text{BR}(N_i \rightarrow Z \nu_\ell + Z \overline{\nu_\ell} ) &=& \text{BR}(N_i \rightarrow h \nu_\ell + h \overline{\nu_\ell} ) = 25\%. \end{eqnarray}
- Helicity dependence of LNV and LNC decay rates of Majorana neutrino decays was validated in [ 7 ]
Studies that have used the above model files
Please email to update this space.
- Golling, et al., Physics at a 100 TeV pp collider: beyond the Standard Model phenomena, arXiv:1606.00947
- CMS, Search for heavy neutral leptons in events with three charged leptons in proton-proton collisions at 13 TeV, arXiv:1802.02965 [hep-ex]
- CMS, Search for heavy Majorana neutrinos in same-sign dilepton channels in proton-proton collisions at 13 TeV, arXiv:1806.10905 [hep-ex]
References
- For studies of Majorana neutrinos, please consider citing [ 6 ] for the Lagrangian and [ 1-2 ] for the Majorana FR/UFO files.
- For studies of heavy Dirac neutrinos, please also consider citing [ 4 ].
- For studies of heavy neutrinos in vSMEFT, please consider citing [ 5 ].
[1] D. Alva, T. Han, R. Ruiz, Heavy Majorana neutrinos from $W\gamma$
fusion at hadron colliders, JHEP 1502, 072 (2015), arXiv:1411.7305 [hep-ph]
[2] C. Degrande, O. Mattelear, R. Ruiz, J. Turner, Fully-Automated Precision Predictions for Heavy Neutrino Production Mechanisms at Hadron Colliders, PRD 94, 053002 (2016), arXiv:1602.06957 [hep-ph]
[3] S. Pascoli, R. Ruiz and C. Weiland, Safe Jet Vetoes, PLB 786, 106 (2018), arXiv:1805.09335 [hep-ph]
[4] S. Pascoli, R. Ruiz and C. Weiland, Heavy Neutrinos with Dynamic Jet Vetoes: Multilepton Searches at $\sqrt{s}=$
14, 27, and 100 TeV, JHEP 1906, 049 (2019), arXiv:1812.08750 [hep-ph]
[5] V. Cirigliano, W. Dekens, J. de Vries, K. Fuyuto, E. Mereghetti, Leptonic anomalous magnetic moments in $\nu$SMEFT
, arXiv:2105.11462 [hep-ph]
[6] A. Atre, T. Han, S. Pascoli and B. Zhang, The Search for Heavy Majorana Neutrinos, JHEP 0905, 030 (2009), arXiv:0901.3589 [hep-ph]
[7] R. Ruiz, A quantitative study on helicity inversion in Majorana neutrino decays at the LHC, arXiv:2008.01092 [hep-ph]
Attachments (18)
-
heavyN.nb
(7.6 KB
) - added by 9 years ago.
Type I Seesaw Model FR mathematica notebook that loads model file, performs basic sanity checks, and outputs to desired format, e.g., UFO.
-
Seesaw_TypeI.tgz
(9.9 KB
) - added by 9 years ago.
Self-contained Type I Seesaw package. Includes: sm.fr (FR default), diagonalCKM.rst (FR default), massless.rst (FR default), heavyN.fr, heavyN.nb
-
Seesaw_TypeI_UFO.tgz
(14.5 KB
) - added by 9 years ago.
Stand-alone UFO model file: assumes top quark and heavy neutrinos are the only massive fermions + diagonal CKM.
-
Seesaw_TypeI_WithUFO.tgz
(24.5 KB
) - added by 9 years ago.
Same content as Seesaw_TypeI.tgz with Seesaw_TypeI_UFO.tgz
-
heavyN_NLO.nb
(17.2 KB
) - added by 9 years ago.
Mathematica notebook to generate UFO files from heavy neutrino FeynRules model file.
-
SM_HeavyN_NLO_UFO.tgz
(23.0 KB
) - added by 8 years ago.
Stand-alone UFO model file: assumes top quark and heavy neutrinos are the only massive fermions + diagonal CKM.
-
heavyN.fr
(8.9 KB
) - added by 8 years ago.
Heavy neutrino FeynRules model file in Feynman gauge.
-
SM_HeavyN_Files.tgz
(12.4 KB
) - added by 8 years ago.
Self-contained Heavy N at NLO package. Includes: sm.fr (FR default), diagonalCKM.rst (FR default), massless.rst (FR default), heavyN.fr, heavyN_NLO.nb
-
SM_HeavyN_FilesWithUFO.tgz
(35.4 KB
) - added by 8 years ago.
Combination of SM_HeavyN_Files.tgz and SM_HeavyN_NLO_UFO.tgz
-
SM_HeavyN_Dirac_LO_UFO.tgz
(17.5 KB
) - added by 6 years ago.
UFO for LO SM_HeavyN_Dirac
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SM_HeavyN_Dirac_NLO_UFO.tgz
(24.5 KB
) - added by 6 years ago.
UFO for NLO SM_HeavyN_Dirac
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heavyN_Dirac.fr
(9.0 KB
) - added by 6 years ago.
Model flie for HeavyN_Dirac
-
SM_HeavyN_Dirac_CKM_Masses_LO_UFO.tgz
(21.8 KB
) - added by 6 years ago.
UFO for HeavyN Dirac model, at LO in QCD, with CKM element, and all particle masses
-
SM_HeavyN_CKM_AllMasses_LO.tgz
(23.0 KB
) - added by 4 years ago.
UFO for HeavyN Majorana model, at LO in QCD, with CKM element, and all particle masses (updated 2020_0813: remove double counting of FeynRules))
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heavyN_vSMEFTdim6.fr
(15.6 KB
) - added by 3 years ago.
.fr file for SM_HeavyN_vSMEFTdim6 (updated to include arXiv info)
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SM_HeavyN_vSMEFTdim6_NLO.tgz
(24.7 KB
) - added by 3 years ago.
Stand-alone NLO UFO for SM_HeavyN_vSMEFTdim6 (updated to include arXiv info)
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SM_HeavyN_vSMEFTdim6_XLO.tgz
(19.2 KB
) - added by 3 years ago.
Stand-alone LO UFO for SM_HeavyN_vSMEFTdim6 (updated to include arXiv info)
-
SM_HeavyN_Gen3Mass_NLO.tgz
(24.9 KB
) - added by 3 years ago.
Stand-alone NLO UFO model file: assumes Gen III (t,b,tau) and heavy neutrinos are massive + diagonal CKM + tau width
Download all attachments as: .zip