Interpretations of the ATLAS Diboson Anomaly

Recently, the ATLAS Collaboration recorded an interesting anomaly in diboson production with excesses at the diboson invariant mass around 2 TeV in boosted jets of all the $WZ$, $W^+ W^-$, and $ZZ$ channels. We offer a theoretical interpretation of the anomaly using a phenomenological right-handed model with extra $W'$ and $Z'$ bosons. Constraints from narrow total decay widths, dijet cross sections, and $W/Z + H$ production are taken into account. We also comment on a few other possibilities.


I. INTRODUCTION
Recently, the ATLAS Collaboration [1] reported an experimental anomaly in diboson production with apparent excesses in boosted jets of the W + W − , W ± Z, and ZZ channels at around 2 TeV invariant mass of the boson pair. 1 It is intriguing because the excesses are all around 2 TeV. The local excesses are at 3.4, 2.6, and 2.9 σ levels for W Z, W + W − , and ZZ channels, respectively (though the global significance of the discrepancy in the W Z channel is only 2.5σ.) The experiment used the method of jet substructure to discriminate the hadronic decays of the W and Z bosons from the usual QCD dijets. The advantage is that the hadronic decays of W and Z afford much larger branching ratios for more events.
However, the jet masses of the W and Z bosons have large overlaps such that the Z boson may be misinterpreted as a W boson, and vice versa. Nevertheless, we attempt to provide a logical explanation for the anomaly.
The anomaly leads to a logical explanation that there exist some exotic particles in some forms of multiplets or under some symmetries (because they have similar masses) with relatively narrow widths decaying into diboson channels. In this note, we propose a phenomenological left-right model that consists of an extra SU (2) R gauge group with (W , Z ) bosons. Initially, we first perform phenomenological studies of the W and Z boson with respect to the data separately. At the end, we shall give a more unified picture of the W and Z bosons coming from a single SU (2) R group.
The W boson couples to the right-handed fermions with a strength g R , which need not be the same as the weak coupling g. The W boson can then be produced via qq annihilation.
Since the W boson is at 2 TeV, the production is mainly via valence quarks and so we anticipate the production cross section of W + is roughly two times as large as the W − cross section at the LHC. The W boson can mix with the SM W boson via a mixing angle, say, sin φ w so that the W boson can decay into W Z with a mixing-angle suppression and right-handed fermions. We shall show that the W decay into W Z dominates if the mixing angle is larger than 10 −2 . Therefore, it can explain the excess in the W Z channel without violating the leptonic cross sections [3][4][5][6] and the dijet-mass search at the LHC [7,8].
The discussion of the Z boson follows closely that of the W boson. It is produced via qq annihilation with a coupling strength g R . The Z boson mixes with the SM Z boson via 1 The CMS Collaboration also saw a moderate excess around 2 TeV in the boosted jets of W + W − , W ± Z, and ZZ [2].
another mixing angle φ z , and then decays into W + W − to explain the excess in the W + W − channel. We adopt a simplified form that the Z only couples to the right-handed fermions, though in general it couples to both left-and right-handed fermions.
There are, in general, a few important constraints that restrict the form the W and Z models: (i) electroweak (EW) precision measurements, (ii) leptonic decays of W and Z , and (iii) dijet production cross sections, plus W H and ZH production that are specific to the current work. The EW precision constraints mainly come from the deviations in the properties of the observed W and Z bosons through the mixings between W and W bosons, and between Z and Z bosons. The measured properties of the W boson restrict the mixing angle between the W and W boson to be φ w < ∼ 1.3 × 10 −2 [9], which is the approximate size of the mixing that is required to explain the diboson anomaly. On the other hand, the constraint on the mixing angle between the Z and Z bosons is much stronger. The updated limits for various Z models, in which the Z boson has direct couplings to SM particles, are of order 10 −3 . This is somewhat smaller than the values required to explain the diboson anomaly. One possibility to relax this constraint is to employ the leptophobic Z model, which is achievable in a number of GUT models [10]. In such a case, the constraint on the mixing angle can be relaxed to 8 × 10 −3 [10], which is close to the value required to explain the diboson anomaly. Therefore, we shall employ the leptophobic Z model in this work.
Furthermore, in the leptonic decays of W + → e + ν R we assume the right-handed neutrinos are heavy enough that the leptonic decays of the W boson are also closed.
Note that we take the excess in the ZZ channel as either a fluctuation or the misinterpretation because of the overlap between the W and Z dijets. On the theory side, it is very difficult to have a spin-1 particle to decay significantly into ZZ, e.g., the Z boson [11,12] or a techni-rho meson. It is possible to have a spin-0 Higgs-like boson to decay into ZZ. However, we found that the production cross section for a 2 TeV Higgs-like boson via gluon fusion is too small to explain the excess in the ZZ channel. Therefore, we take the liberty to ignore the excess in the ZZ channel.
The organization of this note is as follows. In the next section, we describe the interactions of the W and Z bosons, and mixing with the SM W and Z bosons. In Sec. III, we calculate the dijet cross sections to compare with the most updated limits from ATLAS [7] and CMS [8]. In Sec. IV, we calculate the cross sections of pp → W → W Z and pp → Z → W + W − and compare to the ATLAS data. In Sec. V, we give a more unified picture that the W and Z bosons come from a single SU (2) R . We conclude in Sec. VI.
At the last stage of this work, the authors came across Ref. [13] with a similar idea, and Refs. [14,15] in the framework of strong dynamics. There are some existing constraints in literature for models with extra SU (2) [16], especially the dilepton constraint from the LHC experiments. We shall consider the dilepton constraint, as well as the dijet constraint using the most recent data from the LHC.

II. INTERACTIONS OF THE W AND Z BOSONS AND THEIR DECAYS
A. The W boson The extra W 2 boson arises from the right-hand SU (2) R . The right-handed fermions are arranged in isospin doublet, e.g, (u R , d R ) T , (ν R , e R ) T , where ν R is the right-handed neutrino.
The interactions of the W 2 with fermions are given by where P L,R = (1 ∓ γ 5 )/2 and g R is the coupling strength, which need not be the same as the left-handed coupling g but should be of a similar size. The W 1 and W 2 denote the interaction eigenstates, which rotate into the mass eigenstates W and W via a mixing angle φ w (W then represents the observed W boson at 80.4 GeV and the W is the hypothetical 2 TeV Current EW constraints on the W -W mixing angle mainly come from modifying the properties of the observed W boson. The measurements put a limit about 1.3 × 10 −2 [9] on the mixing angle φ w , which is more or less consistent with the values that we use in this study (see Fig. 4). We shall show that such a small mixing angle of order O(10 −2 ) is enough to explain the narrow width of the W bump and the excess in the W Z production cross section.
On the other hand, due to the mixing with the SM W boson, the heavy W couples to W Z with a coupling strength (g cos θ w ) sin φ w , where (g cos θ w ) is the usual coupling constant in the W W Z vertex. On the other hand, the W couples to W H with a full tree-level strength Here we take g R = g. , where p W + ,W − ,Z denote the 4-momenta of the W + , W − , Z bosons going into the vertex and (p W + ), (p W − ), and (p Z ) denote the corresponding polarization 4-vectors.
The partial decay width for W → ff is given by, in massless limit of f, f where N f = 3 (1) for quarks (leptons). Here we also assume that the right-handed neutrinos are so heavy that W + → e + ν R is kinematically not allowed. Therefore, the leptonic decay 2 The mixing angle φ w between the W and W originates from the off-diagonal mass matrix entry, which also gives the tree-level unsuppressed coupling for W -W -H. We found that φ w scales as M 2 W /M 2 W and the coupling for W W H scales as modes of W are closed. The partial width into W Z is [17] It is easy to see that in the W W Z vertex in Eq. (3) the momentum-dependent parts will get enhancement at high energy. Another decay channel of W is W → W H, the partial width of which is given by Note that due to the Equivalence Theorem, In order to avoid a too-broad resonance structure for the W boson we require to be less than one-tenth of the W mass. We show in Fig. 1 the total width of the W boson versus the sine of the mixing angle, and the corresponding branching ratios. Note that the W + H appears to be the same as W + Z in the figure. It is clear that the total decay width grows with sin φ w rapidly. Therefore, the requirement of Γ tot (W ) < ∼ M W /10 gives We show in Table I a few representative values of sin φ w for the partial widths into W Z or W H, and ff and the total width of the W boson. Here we assume g R = g for simplicity.
Note that the total width has only very mild dependence on g R .

B. The Z boson
We repeat the exercise for the Z boson. The interactions of the Z 2 with the SM fermions are given by Case The SM Z 1 boson mixes with Z 2 via a mixing angle φ z into the mass eigenstates Z and Z : Unlike the W -W mixing the EW constraints for the Z-Z mixing angle are much stronger, because of all the precision measurements at LEP. The updated limit for various Z models, in which the Z boson has direct couplings to SM particles, is of order 10 −3 [10]. This is somewhat smaller than the values required to explain the diboson anomaly. One possibility to relax this constraint is to employ the leptophobic Z model, which is achievable in GUT models. In such a case, the constraint on the mixing angle can be relaxed to 8 × 10 −3 [10], which is not far from the values required to solve the diboson anomaly. We shall therefore assume leptophobic couplings of the Z boson. When the mixing angle is of that small size, the Z boson has a narrow width.
The Z boson then couples with a strength proportional to g f,r/l to the SM quarks, but at a strength suppressed by the mixing angle sin φ z to the W + W − . However, the Z boson couples to ZH with a full tree-level strength for a reason similar to the W boson. Now we can write down the relevant vertices of the Z used in this work taking cos φ z 1: The partial widths into ff , W + W − , and ZH are given by In the high energy limit, Γ(Z → W + W − ) Γ(Z → ZH). The total decay width of the Z boson is obtained by summing all the above partial widths. We show the total decay width and partial widths of the Z boson in Fig. 2. The requirement for Γ Z /M Z < 0.1 implies FIG. 2. Total decay width and partial widths of the 2 TeV Z boson versus the sine of the mixing angle φ z . Note that the ZH appears to be the same as W + W − in the figure. Here we take g R = g for simplicity.
Note that the couplings of Z 2 to quarks are model dependent (leptonic couplings are zero). We use the simplified form g f,l = 0, g f,r = g R T 3,f for the interactions according to the right-handed current T The mixing between the Z and Z bosons can be generated through a Higgs boson charged under both symmetries. The mixing is given by φ z = C(g R /g)(M V /M V ) 2 [18], which C can be a definite number or spanned over a range depending on the ratios of the Higgs VEVs.
The mixing angle that we find in this work is about 2 × 10 −3 − 10 −2 and is mostly consistent with the natural value.

III. LIMITS FROM DIJET PRODUCTION AND OTHERS
Since we have assumed the leptophobic Z model and that the right-handed neutrinos are too heavy for W + → e + ν R to occur, the constraints from leptonic cross sections [3][4][5][6] can be ignored. In the following, we first consider the constraints coming from dijet production via σ(W )×B(W → jj) (and similarly for Z ). Both the ATLAS [7] and CMS [8] have searched for resonances decaying into dijets. They pose limits on the current phenomenological W and Z model. We calculate pp → W ± → jj including the width effect and show the production cross sections in Fig. 3, in which we choose g R = 0.6. The acceptance factor A for each experiment is read off from the report of ATLAS and CMS. It is easy to see from both panels that when sin φ w > ∼ 5 × 10 −3 the dijet production cross section at M W = 2 TeV is safe from the experimental limits. As g R further increases, the lower limit on sin φ w increases, as shown in Fig. 4. Note that for g R < ∼ 0.5 there is no lower limit on sin φ w . The Z production cross sections are roughly one half of the W for the same mass of 2 TeV.
Yet, there is another constraint mentioned in Ref. [13]: Both ATLAS [19] and CMS [20] searched for a resonance that decays into a W/Z boson and We summarize in Fig. 4 the allowed parameter space of g R versus sin φ w for the 2 TeV W boson under the following constraints: [19,20].
Similarly, the allowed parameter space in g R versus sin φ z for the Z boson with the following constraints is shown in Fig. 5. [19,20].
As seen in both figures the dijet cross section rules out large values of g R while the narrow width requires sin φ w < ∼ 10 −2 . The overlapping region easily satisfies the W Z/W W and W H/ZH upper limits. As we shall discuss the signal cross sections in the next section, the signal cross section for σ(W ) × B(W → W Z) is of order 5 − 10 fb while that for We show the band of 5 − 10 fb cross sections onto the Fig. 4. The sweet spot is the strip obtained by overlapping the allowed region and the band of 5 − 10 fb. While in Fig. 5 we show the band of 2 − 5 fb with a cyan curve at 3 fb, because of the addition constraint σ(Z ) × B(Z → W + W − ) < 3 fb via semileptonic mode [21].
The favorable region of parameter space in g R versus sin φ w is shown in Fig. 4. We can pick a point in the sweet spot to account for the excess observed in the W Z channel.
From the ATLAS report, the number of excess events is about 8 − 9 events around the 2 TeV peak. The selection efficiency for event topology and boson-tagging requirements is about 13% for a 2 TeV W boson [1]. With a luminosity of 20.3 fb −1 it converts to σ(W ) × B(W → W Z) ≈ 6 − 7 fb (here we take the hadronic branching ratio of a W boson or a Z boson to be 0.7).
In Fig. 4, we show the band of the σ(W ) × B(W → W Z) = 5 − 10 fb. The sweet spot is the strip obtained by overlapping the allowed region and the band of "W Z : 5 − 10 fb".
Let us pick a couple of representative points: (i) sin φ w = 3 × 10 −3 and g R = 0.4 (small mixing but large g R ), and (ii) sin φ w = 1.3 × 10 −2 and g R = 0.2 (large mixing but small g R ).
The mixing angle for the second point is at the upper limit allowed by the EW constraint.
Then we calculate σ(pp → W ± → W ± Z) including the width effect, and add to the dijet background shown in the ATLAS report [1]. We show the sum of the resonance peak and the dijet background in the left panel of Fig. 6. Such a resonance contribution can explain the excess in the W Z channel. We can see that with small mixing but large g R (the cyan histograms) the width of the 2 TeV resonance is narrower while that with large mixing but small g R (the red histograms) is broader. Both choices can account for the data points within the uncertainties.
We repeat the exercise for the Z . The number of excess events is about 7 − 8 events around the 2 TeV peak. The selection efficiency is about the same as the W . It eventually converts to σ(Z ) × B(Z → W + W − ) ≈ 5 − 6 fb. However, due to a recent search [21] using semileptonic decay mode, the 95% CL limit on σ(Z ) × B(Z → W + W − ) < 3 fb. Although there is a slight inconsistency, we pick a couple of representative points such that each gives a cross section about 3 fb: (i) sin φ z = 2.28 × 10 −3 and g R = 0.4, and (ii) sin φ z = 8 × 10 −3 and g R = 0.18 from the sweet spot of Fig. 5. Note that the mixing angle of the second point is at the upper limit allowed by the EW constraint. We show the sum of the resonance peak and the dijet background in the right panel of Fig. 6. Such a resonance contribution can roughly explain the excess in the W + W − channel within uncertainty. used. The dijet backgrounds are given in the ATLAS report [1].
Here we show that it is possible to have unified W and Z bosons in a model with an additional SU (2) symmetry, and it will approach the models of W and Z that we used in Sec. II, III and IV. We follow closely the discussion in a couple of recent works addressing the same anomaly [22]. We start with a popular scenario based on the symmetry breaking pattern from the gauge group SU (2) 1 × SU (2) 2 × U (1) X (with gauge coupling g, g 2 , g X respectively), which is first broken into a lower symmetry SU (2) 1 × U (1) Y at the scale above TeV, and then broken again at the electroweak scale [23]. The intermediate symmetry is just the SM gauge group SU (2) L × U (1) Y . The SM hypercharge convention is fixed by 3 + T 3 + Y X 2 . We choose the leptophobic version of the model such that the right-handed u R and d R quarks are arranged in doublet of the SU (2) 2 while the ν R and e R as singlets of the SU (2) 2 .
The assignment of T and Y X /2 for the right-handed fermions are The first step of symmetry breaking at TeV scale can occur via a Higgs doublet Φ ∼ The gauge field B µ of the U (1) X and the W 3 µ of the SU (2) 2 are rotated by angle φ into the B µ of the U (1) Y and the Z boson: while the second step is the usual breaking of the EW symmetry by another Higgs doublet with a VEV v. In order to obtain the coupling of the B µ the same as the SM hypercharge g 1 Y /2 = (e/ cos θ w )Y /2 in the first step of symmetry breaking, we require The W and Z bosons obtain masses as where x ≡ u 2 /v 2 is very large. Therefore, in leading order M W ≈ M Z if cos φ ≈ 1. This is exactly the limit that we want to pursue, and we shall show that the couplings of the W and Z to fermions will approach the values that we used in the analysis. than 0.3 given g 2 < ∼ 1. In the limit of x being large, the left-handed and right-handed couplings of the W boson to SM fermions become [22] g which is exactly the same as the W interaction in Eq. (1) with g 2 = g R . Similarly, in the limit of large x and small sin φ, the left-handed and right-handed couplings of the Z boson to SM fermions become [22] g f,l −→ g 2 cos φ (T Note that the leptonic couplings g f = ,l/r are suppressed by sin 2 φ and also because its T (2) 3 = 0. The left-handed couplings g q,l of quarks are also suppressed by sin 2 φ. Therefore, only the right-handed couplings of quarks are left unsuppressed, which is close to what we used in the analysis of Z with g 2 = g R in previous sections. Therefore, in the limit of large x we have more or less achieved the leptophobic scenario with W and Z bosons having a similar mass at 2 TeV and couplings to right-handed quarks only.

VI. DISCUSSION
We have considered a phenomenological SU (2)  bosons, respectively, as well as qualitatively the EW precision constraints on the parameter space of g R and the mixing angles φ w and φ z . We have found a sweet spot that satisfies all the constraints, and there exists a viable region that can explain the excess in the W Z and W + W − channels, respectively. The size of the mixing angle is φ w , φ z ≈ 3 × 10 −3 − 10 −2 and the size of the coupling g R ≈ 0.2 − 0.5.
We offer comments on our findings and other possibilities as follows.
1. The production of W Z and W W via W and Z bosons receives a large enhancement due to the longitudinal polarization of the W and Z boson ( µ L (W/Z) ∼ p µ /M W/Z ). If each boson-jet system (which contains 2 closely separated jets) is boosted back to the rest frame of the W/Z boson and the angle made by the jet is measured, one may be able to tell the polarization of the W/Z boson.
2. Another important channel to check is the semileptonic decays of the W and Z bosons, i.e., one boson decays leptonically while the other hadronically. Though the event rates will be lowered, the W or Z peak can be easier distinguished.
3. As we have mentioned that it is very difficult to have a spin-1 boson to decay into ZZ at tree level. There are only two effective operators describing such vertex [11], one of which may be induced by anomaly associated with the extra U (1) while the other must be CP violating. The logical choice is spin-0 or spin-2. However, the production of spin-0 boson, just like the SM Higgs boson, has to go through gg fusion or W W fusion. The production cross sections are too small or the total decay width of the boson is too broad. The spin-2 boson, e.g, the graviton Kaluza-Klein state of the Randall-Sundrum model, can decay into W W and ZZ, but in the ATLAS report [1] it was shown that the production rate of the spin-2 graviton is somewhat too small to explain the anomaly. 4. Another possibility is an extended Higgs sector. It is well-known that in models with extra Higgs doublets the charged Higgs cannot couple to W Z at tree-level. It has to go beyond the doublet to e.g. triplet models. One viable triplet model is the Georgi- Machacek model [24] that contains neutral, singly-charged, and doubly-charged Higgs bosons [25]. The excess in W W channel did not distinguish between W + W − and W ± W ± . In particular, the doubly and singly charged H ++ 5 and H + 5 can be copiously produced via vector-boson fusion for Higgs-boson mass at 2 TeV, but however the width of the bosons are too broad to be consistent [26]. 5. Another alternative is the strong dynamics [15], e.g., technicolor models. For example, a neutral ρ 0 T C of 2 TeV can decay into W + W − while a charged ρ ± T C of 2 TeV can decay into W ± Z.