The Good left-right symmetry, the bad correlated statistical fluke and the ugly systematic error

The Good
I have already reported almost one year ago a putative sign of new physics beyond the Standard Model (SM) at the TeV scale. Since then a few other connected anomalies have shown up providing some momentum for phenomenological investigations. The physicist and excellent blogger Jon Butterworth was recently talking of a definite, intriguing and exciting case of “watch this space”. In this post I would like to share some readings to provide an overview of the current situation:

Recently... a few deviations from the SM predictions have been reported by the ATLAS and CMS Collaborations in invariant mass distributions near 2 TeV:
  • 1) a 3.4σ excess at ∼2 TeV in the ATLAS search [1] for a W' boson decaying into WZ→JJ, where J stands for a wide jet formed by the two nearly collinear jets produced in the decays of a boosted W or Z boson. The mass range with significance above 2σ is ∼1.9–2.1 TeV; the global significance is 2.5σ. A CMS search [2] for JJ resonances, without distinguishing between the W- and Z-tagged jets, has a 1.4σ excess at ∼1.9 TeV. 
  • 2) a 2.8σ excess in the 1.8–2.2 TeV bin in the CMS search [3] for a W0 and a heavy “right-handed” neutrino, NR, through the W'→NRe→eejj process. 
  • 3) a 2.2σ excess in the 1.8–1.9 TeV bin in the CMS search [4] for W'→Wh0, where the SM Higgs boson, h0, is highly boosted and decays into bb̅, while W→ν. 
  • 4) a ∼2σ excess at ∼1.8 TeV in the CMS dijet resonance search [5]. The ATLAS search [6] in the same channel has yielded only a 1σ excess at 1.8 TeV. 

Although none of these deviations is significant enough to indicate a new phenomenon, it behooves us to inquire whether a self-consistent theory may explain all of them. Here we construct a renormalizable theory that explains quantitatively these deviations, and derive its predictions for signals that can be probed in Run 2 of the LHC. 

The deviations showed up in searches for a W' boson but there are several theoretical and experimental hurdles that need to be overcome before a particle of mass near 2 TeV can be inferred. The eejj excess suggests that the W' boson couples to the right-handed quarks, as in the popular left-right symmetric models [7]. However, those models predict that NR has a Majorana mass, so that the number of events with same-sign lepton pairs should be approximately equal to that for opposite-sign lepton pairs [8]. This is ruled out by the CMS excess, which consists almost entirely of e+e- pairs. Thus, we need to modify the left-right symmetric models in order to allow a TeV-scale Dirac mass for NR (alternatively, NR’s with CP violating mixing must be  highly{non} degenerate [9]). 
 Another issue is that all gauge extensions of the SM that include a W' also include a Z' boson. If that Z' couples to the SM leptons, as in left-right symmetric models, then the dilepton resonance searches force the Z' to be significantly heavier than the W'. This constrains the extended Higgs sector responsible for their masses...  
The W' considered here does not directly couple to left-handed leptons, implying highly suppressed W' decays into SM ν pairs (due to the small W−W' mixing). In order to fit the CMS eejj excess, and to avoid large flavor-changing effects, we assume W' coupling to leptons... with the heavy right-handed neutrinos (NeR,NµR,NτR) being part of three vector-like fermions with Dirac masses. Since the CMS µµjj search [3] has not yielded deviations from the SM, the Nµ mass must satisfy mNµ >MW'  ... 
In order for NτR to acquire a Dirac mass we introduce a vector-like fermion ψ=(ψNτ)T transforming as (2,+1) under SU(2)R×U(1)B-L. Its ψNL component can become the Dirac partner of NτR. To see that, let us first describe a simple Higgs sector: an SU(2)R triplet scalar T breaks SU(2)R×U(1)B-L to U(1)Y giving the bulk of MW' and MZ' , and a bi-doublet scalar Σ breaks SU(2)L×U(1)Y→ U(1)Q inducing a small mixing between the charged gauge bosons. For MW'MW , Σ consists of two SU(2) Higgs doublets, which break the electroweak symmetry. The SM Higgs does not mix with other scalars in the alignment limit, and the other charged and neutral scalars could be at the TeV scale as they are also charged under SU(2)R. 
Conclusions.—The W' model presented here appears to be a viable description of the small mass peaks near 2 TeV observed in at least five channels at the LHC. Definitive tests of this model will be performed in several W' decay channels in Run 2 of the LHC. Assuming an SU(2)L×SU(2)R×U(1)B-L gauge origin of the W' , we predict the existence of a Z' boson of mass below 4.5 TeV with production rates shown in Fig.2 {below}. Our renormalizable theory includes Dirac masses for the right-handed neutrinos.

(Submitted on 22 Jun 2015 (v1), last revised 25 Jun 2015 (this version, v2))

* I propose a correction to this statement that seems to me wrong according to my reading of the quoted work.

The Bad
The statistical significance of the signal for new physics is still pretty low. Here is a brief reminder about the general issue of nσ claims by an expert in the field:

It has become the convention in Particle Physics that in order to claim a discovery of some form of New Physics, the chance of just the background having a statistical fluctuation at least as large as the observed effect is equivalent to the area beyond 5σ in one tail of a normalised Gaussian distribution, or smaller...
The traditional arguments for the 5σ criterion are: 
  • History: In the past there have been many ‘phenomena’ that corresponded to 3 or 4σ effects that have gone away when more data were collected. The 5σ criterion is designed to reduce the number of such false claims...
  • Look Elsewhere Effect (LEE): The significance of an interesting peak in a mass spectrum is defined in terms of the probability of a background fluctuation producing an effect at least as large as the one actually seen. If the background fluctuation is required to be at the observed mass, this probability is called the local p-value. If, however, the restriction on the location of the background fluctuation is removed, this is instead the global p-value, and is larger than the local one because of the LEE...
  • Subconscious Bayes’ Factor: When searching for a discovery, the data statistic that is used to discriminate between just background (known as the null hypthesis H0) and ‘background plus signal’ (H1) is often the likelihood ratio L1/L0 for the two hypotheses; and the 5σ criterion is applied to the observed value of this ratio, as compared with its expected distribution assuming just background. However, more relevant to the discovery claim is the ratio of probabilities for the two hypotheses... The above argument is clearly a Bayesian application of Bayes Theorem, while analyses in Particle Physics usually have a more frequentist flavour. Nevertheless, this type of reasoning does and should play a role in requiring a high standard of evidence before we reject well-established theories. There is sense to the oft-quoted maxim ‘Extraordinary claims require extraordinary evidence’...
  • Systematics: It is in general more difficult to estimate systematic uncertainties than statistical ones. Thus a nσ effect in an analysis where the statistical errors are dominant may be more convincing that one where the nσ claim is dominated by systematics errors. Thus in the latter case, a 5σ claim should be reduced to merely 2.5σ if the systematic uncertainties had been underestimated by a factor of 2; this corresponds to the p-value increasing from 3×10-7 by a dramatic factor of 2×104. The current 5σ criterion is partially motivated by a cautious approach to somewhat vague estimates of systematic uncertainties...
There are several reasons why it is not sensible to use a uniform criterion of 5σ for all searches for new physics. These include most of the features that we included as supporting the use of the 5σ criterion. 
This can vary enormously from search to search. Some experiments are specifically designed to measure one parameter, which is sensitive to new physics, while others use general purpose detectors which can produce a whole range of potentially interesting results, and hence have a larger danger of a statistical fluctuation somewhere in the background. An example of an experiment with an enormous LEE is the search for gravitational waves; these can have a variety of different signatures, occur at any time and with a wide range of frequencies, durations, etc...
(Submitted on 4 Oct 2013)

Of course the former article does not deal with the specific issue how many σ should be required for the discovery of W', Z' gauge bosons or NR  right-handed neutrinos. In so far as these particles are simply embedded in a pretty minimal gauge extension of the standard model without going out of the non-susy renormalizable relativistic quantum field theory on 4D Minkowski space-time - the successful paradigm of high energy physics since 50 fifty years - one can expect the same usual 5σ rules... 

The Ugly
To finish with, le's have a look on an intriguing possible systematic error that could jeopardize the exciting interpretation of some of these reported LHC1 anomalies:

The ATLAS Collaboration has recently observed a localised excess in the invariant mass distribution of pairs of fat jets, hereafter denoted by J, around mJJ≃2 TeV [1]. Fat jets can be produced in the hadronic decay of boosted bosons V=W,Z, where the two quarks from the boson decay merge into a single jet. Using jet substructure analyses, the fat jets are tagged as resulting from a boson decay. In addition, in ref. [1] the jets J are identified as W or Z bosons if the jet mass mJ satisfies |mJ−MW | ≤ 13 GeV or |mJ−MZ | ≤ 13 GeV. The excess in the mJJ spectrum appears for W Z, ZZ and WW selections, with statistical significances of 3.4 σ, 2.9 σ and 2.6 σ, respectively [Notice that J can be simultaneously tagged as W and Z with these criteria, as the mass windows for W and Z tagging partially overlap. This indicates, in particular, that a W Z signal can yield significant excesses in the WW and ZZ selections too]. These three channels are not independent and some events fall into two or even the three above categories. A statistical combination of the three channels must take this fact into account, and has not been yet performed.
... it is very unlikely that [HH, WH or ZH resonance signals] could contribute significantly to the JJ excess with a fat dijet selection optimised for VV production [V=W or Z] . Additionally, one expects relations between VV and VH decay fractions of heavy resonances in definite models [13]. All this overwhelming set of related SM-like measurements has motivated the caution by the ATLAS Collaboration regarding this excess, but it has not discouraged early interpretations as new diboson resonances of technicolour models [14,15]. While ref. [14] only takes into account the limit on the production of WZ resonances from the fully leptonic channel (the weakest one), ref. [15] attributes the tension among the searches in different W, Z decay channels to statistics. Other W′/Z′ interpretations [16] only focus on the JJ excess overlooking the null results obtained in the other decay modes of the gauge boson pair 
Statistical fluctuations aside, experimental data seem to disfavour the possibility that the ATLAS JJ excess results from a diboson resonance. We are then led to consider that, it this excess is real, it might be due to something different that looks as a diboson peak due to the kinematical selection applied to reduce SM backgrounds. As we will show in this paper, a requirement on transverse momenta applied in ref. [1] shapes certain resonant VVX signals, with X an extra particle, making them look like a VV resonance. Such a requirement is not used by the corresponding analysis of the JJ final state by the CMS Collaboration [5], nor in the analysis of semi-leptonic final states... 
A heavy resonance decaying into two massive gauge bosons plus an extra particle might explain the peak-shaped excess in the ATLAS diboson resonance search [1] and the absence of such peaks in semi-leptonic channels [2,3,4], nor in the CMS dijet analysis [5]. Simple tests of this hypothesis could be performed by removing the transverse momentum balance requirement in the ATLAS dijet analysis—which would make the excess adopt a broader shape—or, conversely, by introducing this requirement in the rest of searches, especially in the CMS fat dijet analysis. 
Among more exotic candidates, the possibility that X is simply the Higgs boson is quite intriguing. If a WZH resonance R is produced with the above estimated cross section, a 12 fb WH signal will result when the Z boson decays invisibly. In the W leptonic decay mode the invariant mass distribution of the WH pair mℓνJ will concentrate around MR, since the invisible Z still contributes to mℓνJ . For the hadronic channel there are two possibilities that correspond to the two topologies in figure 4 {below}. For the cascade decay R→YZ→WZH, the WH invariant mass mJJ  will peak at the Y mass MY < MR, while for R→YW→WZH, mJJ will be broadly distributed below MR... Therefore, for the topology in figure 4 (b) {below}, a peak should manifest in the WH invariant mass distribution in the semi-leptonic channel but not in the fully hadronic one. This is precisely the behaviour suggested by the CMS semi-leptonic [11] and fully hadronic [10] searches for WH resonances: the former does have a 2.2 σ deviation of ∼ 20 fb at 1.8 TeV whereas the latter, more sensitive, only has an excess at the 1 σ level for this mass. Still, one should bear in mind that statistics are not enough to draw any conclusion. 
The possibly common origin of the ATLAS VV and CMS WH excesses—where the slight mass differences can be attributed to the energy resolution—certainly deserves a more detailed study of the boosted jet tagging and mass reconstruction of W ZH signals. Also, one should bear in mind another 2.8 σ excess in final states with two leptons and two jets at an invariant mass of 2 TeV [22], already interpreted as resulting from new W′ or Z′ vector bosons [23,24,25]. Provided the current excesses are confirmed in 13 TeV data, the higher statistics available will allow for exhaustive tests of the various hypotheses of new resonance production
(Submitted on 22 Jun 2015 (v1), last revised 29 Jun 2015 (this version, v2))

//Last edit 1 July 2015.