Examining a right-handed quark mixing matrix with $b$-tags at the LHC

Encouraged by a hint in a search for right-handed $W$ bosons at the LHC, we investigate whether the unitarity of a right-handed quark mixing matrix and the equality of the left- and right-handed quark mixing matrices could be tested at the LHC. We propose a particular test, involving counting the numbers of $b$-tags in the final state, and simulate the test at the event level with Monte-Carlo tools for the forthcoming $\sqrt{s}=13$ TeV LHC run. We find that testing unitarity with 20/fb will be challenging; our test successfully rejects unitarity if the right-handed quark mixing matrix is non-unitary, but only in particular cases. On the other hand, our test may provide the first opportunity to test the unitarity of a right-handed quark mixing matrix and with 3000/fb severely constrains possible departures from unitarity in the latter. We refine our previous work, testing the equality of quark mixing matrices, with full collider simulation. With 20/fb, we are sensitive to mixing angles as small as $30^\circ$, and with 3000/fb, angles as small as $7.5^\circ$, confirming our preliminary analysis. We briefly investigate testing the unitarity of the SM CKM matrix with a similar method by studying semileptonic $t\bar t$ production, concluding that systematics make it particularly difficult.


I. INTRODUCTION
Earlier this year, a small discrepancy in a search for right-handed (RH) W bosons at the LHC [1] lead to renewed interest in left-right symmetric models [2][3][4][5][6][7][8][9][10][11][12]. Left-right symmetric models were first discussed in the seventies [13][14][15][16][17][18][19] during the early years [20][21][22][23] of grand unification theory (GUT). At moderate energies, such models are described by the gauge symmetries SU(2) L × SU(2) R , and a discrete symmetry, such as parity or charge conjugation, playing the role of left-right symmetry, all of which are spontaneously broken at low energy. These symmetries require RH analogues of the Standard Model (SM) W and Z bosons and of the SM neutrinos. In the SM, the flavor structure of left-handed (LH) quark charged interactions is governed by the CKM matrix [24,25]; similarly, in a left-right symmetric model, distinct quark mixing matrices describe the flavor structure of LH and RH quark charged interactions [26,27]. In our previous work [6], we found that future experiments at the LHC could detect discrepancies between the LH and RH quark mixing matrices. In this work, we refine our previous analysis of left-right symmetry with collider simulations and turn our attention to the unitarity of the RH quark mixing matrix. The unitarity of a three-by-three matrix, implies six orthogonality constraints (widely represented by six unitarity triangles) and three normalization constraints. A common strategy for testing the unitarity of the SM CKM matrix (see e.g., Ref. [28][29][30]) is to make separate, precise measurements of CKM matrix elements. For example, the up-type matrix elements, V uq , can be independently extracted from rare decays. The measurements are in excellent agreement with a unitarity normalization constraint [31], |V ud | 2 + |V us | 2 + |V ub | 2 = 0.9999 ± 0.0006.
The normalization constraint predicts that this quantity is exactly one, while the presence of new physics, especially a fourth generation of quarks, might cause a departure from unitarity in the three-by-three CKM matrix.
In this paper we, however, suggest a different strategy for testing the unitarity of a RH quark mixing matrix, if a RH sector is discovered in the future. Because a RH W boson (henceforth W R boson) would have to be heavier than about 2 TeV to have escaped direct detection [32], or about 2 TeV to explain the anomaly in Ref. [1], decays to all SM quarks are kinematically allowed, and even the top quark can be regarded as approximately massless, m t /M W R 0.1. At a proton collider such as the LHC, jets from the decay of a W R boson would be harder than typical SM backgrounds. Thus, we propose that the unitarity of the RH quark mixing matrix could be tested by analyzing a W R boson's branching fractions at the LHC. In light of Ref. [33], in which the authors stress the enormous numbers of W bosons expected to be produced at the LHC, we briefly discuss a similar strategy for the SM CKM matrix in Sec. III.
To parameterize a non-unitary RH quark mixing matrix (see e.g., Ref. [34]), we first write a product of rotations on six planes in the basis of the quark fields (d, s, b, f ) T , where f is a very heavy, undiscovered fourth-generation down-type quark (see e.g., Ref. [35]): This is a unitary four-by-four RH quark mixing matrix. We introduce a heavy fourthgeneration quark only to parameterize our matrix; we do not assume that it exists in Nature.
The non-unitary three-by-three RH quark mixing matrix is found by omitting the row and column corresponding to the fourth-generation quark, All elements of a non-unitary quark mixing matrix parameterized in this manner, V R , and all elements of V R V † R are less than or equal to one in absolute value. We parameterize a non-unitary quark mixing matrix in this restrictive manner because a fourth-generation quark is a realistic scenario in which the full quark mixing matrix is unitary. An arbitrary non-unitary quark mixing matrix corresponds to non-conservation of probability. To give concrete examples of our methodology, for the sake of simplicity, we often assume either that there is an angle governing the mixing between the first three generations and an angle governing the mixing with the fourth generation: or that the mixing angles in Eq. (4) are equal to a universal mixing angle: We consider W R bosons produced at the LHC at √ s = 13 TeV and decaying via the chain The details of our simulations of such a process are forthcoming. The final W R boson is off-shell (indicated by an asterisk). The jets associated with the W * R decay could carry zero, one or two b-tags, resulting in three categories, i.e., three separate counting experiments.
Because we assume that the RH electron neutrino is the lightest RH neutrino, the off-shell W R boson in this decay chain cannot decay leptonically; the hadronic branching fraction for the off-shell W R boson is 100%. If the RH quark mixing matrix is unitary, the relevant branching fractions and cross sections are described by two parameters: the top-bottom RH quark mixing matrix element, and the cross section, which is a function of the W R boson's mass, the RH neutrino masses, the RH gauge coupling, and the RH quark mixing matrix, and well-known SM parameters. The cross section determines the total number of expected W R boson hadronic decays, whereas the top-bottom RH mixing matrix element, |V R tb | 2 , determines the fraction of hadronic decays with zero, one or two b-tags [6]. Thus, there are three independent measurements described by only two parameters and the RH quark mixing matrix may be "over-fitted," implying that we may test its unitarity.
Let us elaborate upon this claim. Neglecting quark masses, the W R boson's branching fractions into two electrons and first-and second-generation quarks, q, or third-generation quarks, t and b, are The choices of g R , M W R , m ν R , and free parameters in V R other than |V R tb | can affect the total number of hadronic W R decays, but cannot affect the relevant branching fractions or the expected numbers of events with zero, one or two b-tags. By fitting those parameters, one can tune the total number of observed events from the chain in Eq. (8), and by fitting |V R tb | 2 , one can tune, say, the ratio of one and two b-tag events. A remaining independent quantity, the number of zero b-tag events, cannot be tuned. Thus, a unitary quark mixing matrix is "over-fitted," and we can check its unitarity through this undetermined quantity. In fact, unitarity results in restrictions in the branching fractions: Our methodology, however, will test forbidden correlations among branching fractions, as well as these trivial restrictions.
If we assume that the RH mixing matrix is equal to the LH CKM matrix, the total number of events from the decay chain in Eq. (8) Our analysis is not sensitive to a lepton mixing matrix (absent in our Lagrangian). The presence of a lepton mixing matrix could suppress our decay chain, reducing the total number of expected events. The latter could, however, be compensated by increasing the RH coupling.
With those matrix elements, we generated events with Pythia [38,39], linked with the Upon those events, we imposed the following selections on the jets and electrons, based upon those in the CMS search in Ref. [1]: Electrons: 1. We vetoed electrons with pseudo-rapidity η > 2.5.
2. We required exactly two electrons (of any charge).
3. We required that the hardest electron had transverse momentum P T > 60 GeV and the second hardest electron had P T > 40 GeV. 4. We required that the invariant mass of the two hardest electrons was M ee > 200 GeV.

2.
We required at least two jets with P T > 40 GeV, and picked the hardest two.
3. We required that the invariant mass of the two hardest electrons and the two hardest jets was M eejj > 600 GeV.
4. We b-tagged b-jets with a probability of = 0.7.

5.
We b-tagged other jets with a probability of ρ = 0.01. 1 By counting the numbers of events that passed our selections, we estimated the selection efficiencies for the W R boson signal. 2 We picked masses and couplings for the RH sector such that the W R boson could explain the small excess observed in events with two electrons and two jets with an invariant mass of about 2 TeV [1, [9][10][11]. In particular, we decoupled tau and muon RH neutrino masses, but set the electron RH neutrino mass, m ν e R = 1 2 M W R , such that W R bosons decayed via electronneutrinos. We imposed M W R = 2 TeV, g R = 1 2 g L and a diagonal RH quark mixing matrix, 1 Although PGS4 permits detailed, momentum dependent tagging algorithms, all such algorithms require that PGS4 is an "oracle" that reveals a jet's true flavor and calculate a probability that a jet is tagged.
We picked a crude algorithm to simplify our analysis, but which is a reasonable approximation to an experimental analysis. 2 We scrutinized events from PGS4 in the LHCO format with a new code, LHCO_reader [42].
If our selections and b-tagging algorithms resulted in no impurities, the matrix would read: If our efficiencies were 100%, the constants in this matrix would be equal to one. Our selection efficiency for V R = V L without a cut on the invariant mass M eejj is about 75%, which compares reasonably with 78% quoted in Ref. [1].
We select the two hardest jets, which we assume originate from the W R boson. There is, however, an appreciable chance that our procedure selects a rogue jet, damaging our purities. A W R boson decaying to a top quark is especially problematic, because the top itself  (16)). In principle, it might be possible to improve our efficiencies with a dedicated analysis.

B. Standard Model backgrounds
The dominant SM backgrounds are dileptonic tt and Drell-Yan, which potentially contaminate each of our b-tags categories due to the imperfections in the b-tagging algorithm.
These processes have substantial cross sections and impair our W R search in the tails of their kinematic distributions, yielding technical difficulties in dedicated simulations. Instead of performing a specific analysis we then choose to model our backgrounds as follows. We first consider a number of background events that after all selections, efficiencies, etc, matches the data-driven prediction in Ref. [1] at √ s = 8 TeV, where the background is estimated from a sideband. In order to account for the different center-of-mass energy considered in our study, we then apply to the quoted number of background events a further rescaling factor given by the ratio or the relevant cross sections calculated at 8 and 13 TeV. In order to give a conservative estimate, we furthermore assumed that all background events result in a b-tag distribution identical to that of tt production. Finally, as a crosscheck, we increased the expected background by a factor of 50%, finding that our conclusions are robust and the sensitivity of our test weakens only by a few degrees.

C. Statistical treatment
We examine the RH quark mixing matrix with a methodology similar to that in our previous work [6], with which we tested the equality of the LH and RH quark mixing matrices.
The statistical treatment is based upon Poisson statistics of three independent counting experiments: the numbers of hadronic W R boson decays resulting in zero, one or two b-tags.
We conduct two similar statistical tests: a test of the equality of the LH and RH quark mixing matrices, and a test of the unitarity of the RH quark mixing matrix.

Testing the unitarity of the right-handed quark mixing matrix
We consider two hypotheses in our statistical test of the unitarity of the RH quark mixing matrix: • The null hypothesis, H 0 : the RH quark mixing matrix is unitary, that is, In this case, the numbers of expected events in each b-tag category resulting from Eq. (8) are described by a cross section, σ, and the top-bottom quark mixing matrix element, |V R tb | 2 .
• The alternative hypothesis, H 1 : the RH quark mixing matrix is non-unitary. In this case, the numbers of expected events are described by a cross section, σ, and the independent elements of the non-unitary RH quark mixing matrix.
To test whether future LHC experiments could reject the null hypothesis that the RH quark mixing matrix is unitary, we construct a log-likelihood ratio (LLR) test statistic and find p-values, assuming that Nature is described by a non-unitary RH quark mixing matrix.
Having picked a particular non-unitary matrix, we: 1. Calculate the expected numbers of signal and background events in the zero, one and two b-tag categories, λ i , with i = 0, 1 or 2. For simplicity, we assume that the production cross section is independent of the quark mixing matrix and always equal to that obtained in Sec. II A. This is because changes in the production cross section due to the quark mixing matrix can be compensated by altering the RH coupling.
2. Simulate a counting experiment, by drawing 1000 samples, {o i }, from Poisson distribu- 3. For each sample, we calculate our LLR for our hypotheses: where the likelihood functions, L, are Poisson distributions and a line above a parameter indicates that the value of that parameter is chosen such that the likelihood is maximized.
In the numerator, V R tb refers to a single element of the quark mixing matrix, whereas in the denominator, V R is the RH quark mixing matrix.
We found the LLR distribution from MC. The p-value is the probability of obtaining such a large LLR by chance, were the null hypothesis true -the area under the right-hand-side tail in the LLR distribution.
4. Lastly, we calculate the median and 68% confidence interval for the p-value, by considering all of our pseudo-experiments.

Testing the equality of left-and right-handed quark mixing matrices
Similarly, we consider two hypotheses in our statistical test of the equality of LH and RH quark mixing matrices: • The null hypothesis, H 0 : the RH quark mixing matrix is equal to the LH quark mixing matrix, that is, In this case, the numbers of expected events in each b-tag category resulting from Eq. (8) are described by a cross section, σ. The fractions of events in each b-tag category are determined by the (fixed) LH quark mixing matrix.
• The alternative hypothesis, H 1 : the RH quark mixing matrix is an arbitrary, non-unitary three-by-three matrix. In this case, the numbers of expected events are described by a cross section, σ, and the independent elements of the non-unitary RH quark mixing matrix.
Assuming that Nature is described by a unitary RH quark mixing matrix, we follow an identical sequence of steps as before, though with a different LLR test statistic: As previously, the likelihood functions, L, are Poisson distributions, and a line above a parameter indicates that the value of that parameter is chosen such that the likelihood is maximized.

III. UNITARITY OF STANDARD MODEL CKM MATRIX
It is possible to extract the SM CKM matrix element V tb from single top production [43] and, if one assumes unitarity, from tt production [44]. We believe that it is, in principle, possible to test the unitarity of the SM CKM matrix and constrain CKM matrix elements at the LHC by analyzing W boson decays. The semileptonic tt process, could be reconstructed with two b-tags, a lepton and a top-tagging algorithm. If the CKM matrix is unitary, the total number of W → qb and W → qq events ought to follow n(qb events) = const. × |V tb | 4 (1 − |V tb | 2 ) and n(qq events) = const. × |V tb | 4 (1 + |V tb | 2 ), whereas if the CKM matrix is not unitary, n(qb events) = const. × R 2 η and n(qq events) = const.
where we define R ≡ |V tb | 2 /(|V ts | 2 + |V td | 2 + |V tb | 2 ) and η ≡ (|V ub | 2 + |V cb | 2 )/(|V ub | 2 + |V cb | 2 + |V ud | 2 + |V us | 2 + |V us | 2 + |V cs | 2 ). These decays are distinguishable in an experiment with a b-tagging algorithm (because W → tb is kinematically forbidden, there is no two b-tag category). With two measurements, the V tb element is over-constrained. We find, however, that regardless of large statistics of about 10 9 W bosons, the test is thwarted by systematic errors in the tt cross section and backgrounds. It may, however, be possible to extract and verify information about the CKM matrix in such an analysis.

IV. RESULTS
Let us recapitulate our goal. Assuming that a W R boson is discovered in the future, we want to know whether a √ s = 13 TeV LHC analysis of W R bosons is sensitive to the unitarity of the RH quark mixing matrix or the equality of the LH and RH quark mixing matrices.
We consider the cases separately, in the following two sections. We initially consider 20/fb of data at √ s = 13 TeV, before investigating the ultimate statistical power of our tests with 3000/fb, although their sensitivities may improve if efficiencies and purities, especially in the two b-tag category, are refined.

A. Unitarity
We plot the potential to exclude V R V † R = 1 against a single universal mixing angle, θ ≡ θ 3 ≡ θ 4 , with 20/fb in Fig. 1. Whereas the upper panel shows the expected fraction of signal events in each b-tag category, in the lower panel we plot the median p-value (solid blue line), as well as its 68% range (filled blue band). If the former drops below 5% (dashed magenta line), in the majority of cases, we could reject the null hypothesis that V R = V L with at least 95% confidence. As we can see, if the mixing angle is greater than about 75 • , a unitary matrix cannot reproduce the fractions of b-tags, because Eq. (14) Testing V R V R =1 with 20/fb Median p-value with 68% range Having considered 20/fb, we now analyze a scenario with 3000/fb of data, corresponding to the entire LHC operation. With such large statistics, the two b-tag category finally helps to split the hypotheses, as we show in Fig. 2, in which we plot Fig. 1 Testing V R V R =1 with 3000/fb Median p-value with 68% range Figure 2: As in Fig. 1, but with 3000/fb. Potential to exclude a unitary RH quark mixing matrix, Z boson resonance (and the RH lepton mixing matrix was assumed or known). As a remark, we parameterized a non-unitary quark mixing matrix as a sub-block of a four-by-four unitary matrix. If we generated RH quark mixing matrices with random entries from zero to one that satisfy the restrictions in Eq. (14), with 3000/fb, we could reject unitarity in about 75% of cases.
Our results for unitarity are sensitive to the mixing angles between the first three generations of RH quarks. In Fig. 3, we plot the median exclusion on the (θ 3 , θ 4 ) plane with 3000/fb. The angle θ 4 parameterizes the departure from unitarity (if θ 4 = 0, the RH quark mixing matrix is unitary) and the angle θ 3 describes the mixing among the SM RH quarks (see Eq. (6)). We see from Fig. 3 that to reject V R V † R = 1 at 95% confidence, we require that the departure from unitarity is substantial, θ 4 75 • or that other mixing angles are moderate, θ 3 Fig. 4) as a function of the mixing angle is flatter than that in FM1, because of the imperfect efficiencies and purities achieved in our collider simulations, but the resulting exclusion is similar. In the upper panel in Fig. 4, we confirm that the 68% range for the p-value shrinks rapidly, as found in our previous work.
In FM1, we also considered 3000/fb, and found that if the mixing angles were as small as about 7.5 • , our analysis could reject V R = V L . We repeat this test in Fig. 5 Median p-value with 68% range Figure 5: As in Fig. 4, but with 3000/fb. Potential to exclude a unitary RH quark mixing matrix, V R = V L , as a function of a universal mixing angle by counting b-tags at the LHC with √ s = 13 TeV, L ∼ 3000/fb. When the p-value falls below 5%, we can reject the null hypothesis, that the RH quark mixing matrix is equal to the LH quark mixing matrix, with 95% confidence. The upper panel shows the expected fractions of b-tags in the case in which V R = V L .

C. Possible improvements with a top-tagging algorithm
Events involving top quarks can be identified by inspecting jet substructure and jet mass.
This is known as top-tagging (see e.g., Ref. [45]). Top-tagging could improve the sensitivity of our analysis by expanding our categories with a t-tag category: Number 0 b-tags and 0 t-tags An event is top-tagged if a combination of objects in that event satisfies a top-tagging algorithm; however, we forbid jets in such combinations from resulting in a further b-tag.
As well as extending our categories, a top-tagger may reduce the contamination between the former. For example, a W R → tb event, which ought to be categorized as a two b-jet event, may contaminate the one b-tag category in our analysis, if, amongst other things, a secondary jet from the top-decay is selected. A top-tagger may reduce contamination in such events because, once a combination of objects is successfully top-tagged, there is no risk of wrongly selecting a jet from a W -boson decay resulting from the top decay, instead of the primary jet from the top decay.
We model the potential impact of a top-tagger by supposing that top-tagging algorithms increase the chances of correctly selecting the primary jet from a top-decay. We keep only three event categories, but assume that contamination between categories is reduced, which we implement with an improved efficiency matrix. We find that improvements in our efficiency matrix yield diminishing returns, at best extending our sensitivity to RH mixing angles by a few degrees.
We could, in principle, perform an analysis with a top-tagger and with the four categories in Eq. (25). The division of the one b-tag category into a b-tag category and a t-tag category might improve sensitivity. However, all RH mixing matrices that are symmetric modulo phases 3 (SMP) make approximately identical predictions for the ratio of the numbers of events in these two categories (the predictions are identical if there are no impurities). Thus, this extra information from the top-tagger might not help to discriminate between a unitary (and thus SMP) RH mixing matrix and an SMP non-unitary RH mixing matrix, as considered in our analysis.

V. CONCLUSIONS
Building upon our earlier work, we proposed methods with which one could examine a RH quark mixing matrix at the LHC at √ s = 13 TeV, if a W R boson with a mass of about 2 TeV, hinted at by experiments at √ s = 8 TeV, were discovered in a final state with two electrons and two jets. Our methods involved counting the numbers of b-tags from a W R boson decay. If the RH quark mixing matrix is unitary, particular relations between the branching fractions are required (other than the trivial condition that branching fractions sum to unity), which could, in principle, be checked.
With 20/fb, we find that the sensitivity of our test is somewhat limited by small statistics in the two b-tag category. One can exclude unitarity, but only for particular structures of RH quark mixing matrix; even maximal mixing with a fourth-generation quark by itself is insufficient. On the other hand, with 3000/fb, our method is able to significantly constrain departures from unitarity in the RH quark mixing matrix. If the anomaly observed in Ref. [1] persists and in the future a W R boson is discovered, our test could be the first check of the unitarity of the RH quark mixing matrix. This would be an important test because rejecting unitarity would cast doubt upon the simplest interpretation of a W R boson. We briefly discussed the application of this method to the unitarity of the LH CKM quark mixing matrix.
Concerning the possible inequality of the LH and RH quark mixing matrices, their difference is statistically significant with 20/fb if the mixing angles in the RH quark mixing matrix are greater than about 30 • and with 3000/fb, we achieve sensitivity to angles as small as about 7.5 • . Our findings with a full MC collider simulation are in agreement with our preliminary analysis FM1 [6]. We suspect that our efficiencies and purities, especially for the two b-tag category, could be further optimized. If the anomaly persists, one challenge might be improving the efficiencies to maximize the insights into the RH quark mixing matrix.