Revealing the electroweak properties of a new scalar resonance

One or more new heavy resonances may be discovered in experiments at the CERN Large Hadron Collider. In order to determine if such a resonance is the long-awaited Higgs boson, it is essential to pin down its spin, CP, and electroweak quantum numbers. Here we describe how to determine what role a newly-discovered neutral CP-even scalar plays in electroweak symmetry breaking, by measuring its relative decay rates into pairs of electroweak vector bosons: WW, ZZ, \gamma\gamma, and Z\gamma. With the data-driven assumption that electroweak symmetry breaking respects a remnant custodial symmetry, we perform a general analysis with operators up to dimension five. Remarkably, only three pure cases and one nontrivial mixed case need to be disambiguated, which can always be done if all four decay modes to electroweak vector bosons can be observed or constrained. We exhibit interesting special cases of Higgs look-alikes with nonstandard decay patterns, including a very suppressed branching to WW or very enhanced branchings to \gamma\gamma and Z\gamma. Even if two vector boson branching fractions conform to Standard Model expectations for a Higgs doublet, measurements of the other two decay modes could unmask a Higgs imposter.


I. INTRODUCTION
Experiments at the Fermilab Tevatron and the CERN Large Hadron Collider are engaged in searches for the Higgs boson, a heavy scalar resonance predicted by the Standard Model (SM). SM Higgs bosons are excitations of the neutral CP -even component of an SU(2) L weak isospin doublet field H carrying unit hypercharge under U(1) Y , whose vacuum expectation value (VEV) v/ √ 2 is responsible for electroweak symmetry breaking (for reviews, see [1,2]).
If one or more new heavy resonances are discovered at the LHC, it will be imperative to pin down their quantum numbers relative to the expected properties of the SM Higgs.
Determination of the spin and CP properties of a new resonance will be challenging, although recent studies indicate that definitive results could be obtained at or around the moment of discovery, if the decay mode to ZZ is observable [3][4][5].
Given a neutral CP -even spin 0 resonance S, one still needs to establish its electroweak quantum numbers in order to reveal any possible connection to electroweak symmetry breaking. This in turn requires information about the couplings between S and pairs of vector bosons, which can be extracted from observations of S decaying via W + W − , ZZ, γγ, or Zγ. To an excellent approximation the couplings of the SM Higgs boson to W W and ZZ derive from the dimension-four Higgs kinetic terms in the SM effective action, and are thus directly related to both the strength of electroweak symmetry breaking and the electroweak quantum numbers of the Higgs field. The couplings of the SM Higgs boson to γγ, Zγ, or a pair of gluons are elegantly derived from the observation that Higgs couplings in the SM are identical to those of a conformal-compensating dilaton in a theory where scale invariance is violated by the trace anomaly [6][7][8][9]. Thus these couplings appear first at dimension five, with coefficients related to SM gauge group beta functions.
In this paper we exhibit a general analysis, up to operators of dimension five, of the relation between the electroweak properties of S and its decay branchings to V 1 V 2 = W W , ZZ, γγ, and Zγ. We ignore decays into two gluons because of the folklore that these are unobservable, and postpone until the end a discussion of extracting complementary information from vector boson fusion production of S [10,11]. Nevertheless, we should emphasize that our analysis only involves the decays of the scalar into electroweak vector bosons, and hence is independent of the production mechanism of the scalar.
A key feature of our analysis is the classification of Higgs look-alikes according to the custodial symmetry SU(2) C . In the SM this global symmetry is the diagonal remnant after electroweak symmetry breaking of an accidental global SU(2) L × SU(2) R , in which SU(2) L and the U(1) Y subgroup of SU(2) R are gauged. Custodial symmetry implies ρ ≡ m 2 W /(m Z c w ) 2 = 1 [12], where c w is the cosine of the weak mixing angle. Experimentally ρ is constrained to be very close to one [13], implying either that the full scalar sector respects SU(2) C , or that there are percent-level cancellations unmotivated by symmetry arguments.
In our analysis we will assume that unbroken SU(2) C is built into the scalar sector.
We consider S arising from one of the neutral CP -even components of arbitrary spin 0 multiplets of SU(2) L × SU(2) R . The case of a singlet under SU(2) L × SU(2) R is special, since then no SV 1 V 2 couplings can appear from operators of dimension four. All other cases can be grouped according to whether the neutral scalar components transform as a singlet or a 5-plet under SU(2) C . Again under the assumption that the full scalar potential respects the custodial symmetry, these three "pure cases" only give rise to one nontrivial mixed case, i.e. when S from a SU(2) L × SU(2) R singlet mixes with another S from the SU(2) C singlet part of a SU(2) L × SU(2) R nonsinglet. Given the framework just described, we are able to enumerate all possible deviations from the SM expectations for decays of a Higgs look-alike into pairs of electroweak bosons.
These deviations are typically quite large, and thus accessible to experiment at the LHC.
Furthermore the deviations exhibit patterns that point towards particular non-SM scenarios.
It would therefore be possible with LHC data to rule out a new scalar resonance as the agent (or the sole agent) of electroweak symmetry breaking. This possibility emphasizes the importance of observing all four V 1 V 2 decay channels at the LHC with maximum sensitivity.
We give examples of Higgs imposters that meet SM expectations for branching fractions into two of the electroweak V 1 V 2 modes, only revealing their ersatz nature in the other two V 1 V 2 decay modes. The approach taken here is complimentary to that in Refs. [3][4][5], where angular correlations and total decay width were used to distinguish Higgs look-alikes. A fully global analysis using all of the available decay and production observables in each channel will of course give superior results to the simple counting experiments described here.
In Sect. II we describe the dimension four couplings of an arbitrary neutral CP -even scalar charged under SU(2) L ×U(1) Y to W W or ZZ; we also describe the general dimension five couplings of a SU(2) L × U(1) Y singlet to two electroweak vector bosons. Sect. III contains the general framework based on custodial symmetry. In Sect. IV we provide general results on the patterns of S → V 1 V 2 branching fractions, as well as discussing some interesting special cases. Further discussion and outlook are in Sect. V, with some general formulae for off-shell decays relegated to an appendix.
In this section we consider scalar couplings with two electroweak gauge bosons where V 1 V 2 = {W W, ZZ, Zγ, γγ}. Such couplings are dictated by the electroweak quantum numbers of the scalar S. We will write down SU(2) L × U(1) Y invariant operators giving rise to the SV 1 V 2 couplings at the leading order. For an electroweak nonsinglet, the leading operator is the kinetic term of the scalar, assuming S receives a VEV, while for the singlet scalar the leading operator starts at dimension five.
For nonsinglet scalars, the leading contribution to the SV 1 V 2 coupling arises from spontaneous breaking of SU(2) L × U(1) Y down to U(1) em via the Higgs mechanism, when S develops a VEV. It is possible to derive the general coupling when there are multiple scalars in arbitrary representations of the SU(2) L group [16,17]. Using the notation φ k for scalars in the complex representations and η i for scalars in the real representations 1 , the kinetic where is the covariant derivative. In the above W a µ and g are the SU(2) L gauge bosons and gauge coupling, respectively, while B µ and g ′ are the U(1) Y gauge boson and gauge coupling. In addition, T a are the SU(2) L generators in the corresponding representation of the scalar, and Y is the hypercharge generator. For complex representations we work in the basis where T 3 and Y are diagonal. After shifting the scalar fields by their VEV's: φ k → φ k + φ k and η i → η i + η i , where the VEV's are normalized as follows 1 A real representation is defined as a real multiplet with integer weak isospin and Y = 0. electroweak symmetry is broken and W and Z bosons become massive. The mass eigenstates are defined as where the sine and cosine of the weak mixing angle are c w = g/ g 2 + g ′2 and s w = g ′ / g 2 + g ′2 , respectively. Notice the unbroken U(1) em leads to the conditions Using T 3 φ k = −Y φ k /2 it is possible to express the mass terms of the W and Z in terms where Y k and Y i are the hypercharges of φ k and η i . Couplings of the real component of the neutral scalar with the W and Z can be read off by the replacement v k → v k (1 + φ 0 k /v k ) and v i →ṽ i (1 + η 0 i /ṽ i ) in the mass terms: Notice that a scalar in a real representation only couples to W W but not ZZ.
Two more popular examples are the real triplet scalar φ and the complex triplet scalar Φ with (T, Y ) = (1, 0) and (T, Y ) = (1, 2), respectively, for which the couplings are We see that the SV 1 V 2 couplings are distinctly different for scalars carrying different electroweak quantum numbers, which would give rise to different patterns of decay branching ratios into two electroweak vector bosons. However, it is well known that φ and Φ individually violate the custodial symmetry and leads to unacceptably large corrections to the ρ parameter unless the VEV is extremely small, on the order of a few GeV [13][14][15].
For a singlet scalar s, the sV 1 V 2 couplings do not come from the Higgs mechanism.
Instead, they originate from the following two dimension-five operators at the leading order: where the singlet s is assumed to be CP -even. We have normalized the dimensionful couplings to the mass of the singlet m s , although in general an unrelated mass scale could enter.
In terms of the mass eigenstate in Eq. (4), the operators become from which we obtain the following couplings: One sees immediately that branching ratios following from these couplings are distinctly different from those coming from the Higgs mechanism. Moreover, the four couplings are controlled by only two unknown coefficients κ 2 and κ 1 . So measurements of any two couplings would allow us to predict the remaining couplings, which, if verified experimentally, would be a striking confirmation of the singlet nature of the scalar resonance.
It is worth commenting that the coefficients κ 2 and κ 1 are related to the one-loop beta functions of SU(2) L and U(1) Y gauge groups, respectively, via the Higgs low-energy theorem [18,19]: In the above the Casmir invariants are defined as while n s is the number of scalars in the complex representation r and n ′ s is the number of scalars charged under U(1) Y . Such a connection has been exploited to compute that partial width of h → gg and h → γγ in the standard model [18,19], as well as to derive the constraints on the Higgs effective couplings [20]. For our purpose such relations serve to demonstrate that the special case of κ 2 = κ 1 , where the ratio of singlet couplings with W W and ZZ coincides with the standard model expectation, in general requires a conspiracy between the two one-loop beta functions to cancel each other. In this case, however, the coupling to γγ is identical to the coupling to ZZ. On the other hand, depending on whether the SU(2) L running is asymptotically free, κ 2 and κ 1 could have either the same or opposite sign, resulting in a reduction (same sign) or enhancement (opposite sign) of the Zγ width relative to ZZ and γγ channels. It is also possible that κ 2 = 0, resulting in a very suppressed decay width into W W . We will discuss further these special cases in Sect. IV.

III. IMPLICATIONS OF CUSTODIAL INVARIANCE
We have seen in the previous section that scalar couplings with two electroweak bosons are uniquely determined by the SU(2) L × U(1) Y quantum number of the scalar involved.
For nonsinglet scalars the leading contribution to the SV 1 V 2 couplings come from the kinetic terms via the Higgs mechanism, which in turn are related to the contribution of each scalar VEV to the masses of the W and Z bosons. However, the ratio of the W and Z masses are measured very precisely and related to the precision electroweak observable which is determined at the tree-level by the structure of the scalar sector in a model. Experimentally ρ is very close to 1 at the percent level [13], which severely constrains the electroweak quantum number of any scalar which develops a VEV.
It has been known for a long time that the Higgs sector in the standard model possesses an accidental global symmetry SU(2) L × SU(2) R , in which the SU(2) L and T 3 R are gauged and identified with the weak isospin and the hypercharge, respectively. After electroweak symmetry breaking the global symmetry is broken down to the diagonal SU (2), which remains unbroken. The unbroken SU (2) is dubbed the custodial symmetry in Ref. [12], where it was shown the relation ρ = 1 is protected by the custodial symmetry SU(2) C . In this section we classify scalar interactions with two electroweak vector bosons according to the SU(2) C quantum number of the scalar. 3 There are two possibilities for the scalar sector of a model to preserve the SU(2) C symmetry. One could find a single irreducible representation of SU(2) L × U(1) Y which realizes ρ = 1. In this case there is only one neutral CP -even scalar and the W and Z obtain masses from a single source, the VEV of the neutral scalar S 0 . From Eqs. (6) and (7) we see the condition to realize this possibility is An obvious solution is the Higgs doublet (T, Y ) = (1/2, 1), beyond which the next simplest case is (T, Y ) = (3, 4) [16]. However, it is clear that, since there is only one source for the masses of W and Z bosons, the SV 1 V 2 couplings are derived by replacing in the mass term, which results in In other words, when there is only a single source for the mass of electroweak bosons, the custodial symmetry uniquely determines the ratio of the scalar couplings to W W and ZZ to be regardless of the SU(2) L ×U(1) Y quantum number of the scalar involved. In the next section we will see that Eq. (22) predicts the ratio of the decay branching fractions into W W and ZZ to be roughly two-to-one, which is the case in the SM with a Higgs doublet.
The second possibility is to consider multiple scalars all contributing to the W and Z masses through the Higgs mechanism in such a way that, although individually the custodial invariance is not respected, the ρ parameter remains 1 due to cancellations between the spectively. Recall that SU(2) L is fully gauged and identified with the weak isospin, while Therefore, all neutral components in the scalar multiplets have T 3 C = 0. On the other hand, unbroken custodial symmetry requires that only SU(2) C singlets are allowed to have a VEV.
In other words, the scalar representation (M L , N R ) must contain a state with T C = 0, where T C is the eigenvalue labeling the quadratic Casmir operator T a C T a C = T C (T C + 1)1 1. Since T C satisfies we conclude that ρ = 1 is possible only when M = N and the scalar must furnish the The trivial representation (1 L , 1 R ) is a singlet scalar under SU(2) L × U(1) Y , which was considered in the previous section. In the following we focus on the non-trivial representations, in which the SV 1 V 2 couplings arise from the Higgs mechanism after the electroweak symmetry breaking. We will represent a scalar Φ N in the (N L , N R ) multiplet in a N × N matrix whose column vectors are N-plets under SU(2) L . The kinetic term of Φ N is where T a are generators of SU (2) in the N-plet representation. When Φ N develops a VEV in a custodially invariant fashion 4 electroweak symmetry breaking occurs and ρ = 1 at the tree-level.
In general various scalars in Φ N could mix with one another and the mass eigenstates do not necessarily have well-defined SU(2) L × U(1) Y quantum numbers. However, it is highly desirable that the scalar potential respects the custodial symmetry so as to be consistent with ρ = 1, which we assume to be the case. Then scalars with different SU(2) C quantum numbers do not mix and all the mass eigenstates have definite SU(2) C quantum numbers, according to which we will proceed to classify the SV 1 V 2 interactions. The (N L , N R ) representation decomposes under the unbroken SU(2) C as Scalars in the (4k + 1)-plet are CP -even and those in the (4k + 3)-plet are CP -odd. We assume no CP -violation in the scalar sector and neglect the CP -odd scalar interactions.
Since we are interested in interactions with two electroweak gauge bosons, it is worth recalling that W a µ and B µ transform as (part of) (3 L , 3 R ) under SU(2) L × SU(2) R . Therefore the only possible SU(2) C quantum numbers of a system of two electroweak gauge bosons are a singlet, a triplet, or a 5-plet, which implies the scalar must also be in one of the above three representations in order to have a non-zero coupling with two electroweak bosons.
We conclude that CP -even SV 1 V 2 interactions are allowed only when the scalar is either a SU(2) C singlet or a 5-plet. This is equivalent to saying two spin-1 objects can only couple to either a spin-0 or a spin-2 object. Interactions of two electroweak bosons with scalars in higher representations of SU(2) C all vanish.
Let's define the the neutral component of a custodial n-plet as H 0 n = h 0 n X 0 n , where h 0 n is the neutral scalar field and X 0 n is a N × N diagonal matrix satisfying 5 [T a T a , X 0 n ] = n(n + 1)X 0 n , [T 3 , X 0 n ] = 0 , Tr(X 0 n X 0 n ) = 1 .
As emphasized already, only h 0 1 is allowed to develop a VEV. From Eq. (diagonal) traceless matrices: The VEV of h 0 1 gives rise to the following masses from the kinetic term of Φ N : which exhibits ρ = 1. It can be verified explicitly that Eqs. (31) and (32) are consistent with Eqs. (6) and (7). Interactions of h 0 n , n = 1, 5, with electroweak bosons can be obtained by setting Φ N = (v/ √ 2)1 1 + H 0 n in Eq. (25): For the custodial singlet, n = 1 and X 0 1 = 1 1/ √ N , we obtain which is a demonstration of the statement that any custodial singlet (apart from the one in the trivial representation (1 L , 1 R )) must have couplings to the W W and ZZ bosons with a fixed ratio as in Eq. (22). On the other hand, since X 0 5 is a traceless diagonal matrix, we have Then the couplings are which turn out to have a ratio g h 0 that is different from the ratio of c 2 w for the custodial singlet h 0 1 . We emphasize that the ratios of the couplings only depend on the SU(2) C quantum numbers, and not on the particular (N L , N R ) representation.
Again we discuss a few examples. The canonical example is the familiar Higgs doublet:  11) and (12). In this case, the SU(2) C quantum numbers are (3 L , 3 R ) = 1 ⊕ 3 ⊕ 5, which contains two CP -even neutral scalars in the singlet and the 5-plet and one CP -odd scalar in the triplet [21]. Our expressions for couplings of the singlet and the 5-plet with W W and ZZ are consistent with those in Refs. [21][22][23]. 6 It is also possible that the scalar sector of a model has multiple neutral scalar particles.
In this case only scalars within the same SU(2) C multiplet are allowed to mix in order to preserve ρ = 1. Then the ratio of the SV 1 V 2 couplings in the mass eigenstate depends only on the SU(2) C quantum number and not on the mixing angle at all, except when there exists an electroweak singlet scalar s which couples to V 1 V 2 through the higher dimensional operators in Eq. (13). In this case, it is necessary to include the loop-induced couplings of h 0 1 with Zγ and γγ since they are in the same order as the sV 1 V 2 couplings. Furthermore, there could be a higher dimensional operator of the form s|D µ Φ N | 2 , with the coefficient κ s /m s , which gives rise to the coupling sV µ 1 V 2 µ in addition to those in Eq. (15). Even so, there are only seven unknown parameters: g h 0 1 W W , g h 0 1 Zγ , g h 0 1 γγ , κ 1 , κ 2 , κ s , and the mixing angle between h 0 1 and s, while one could measure a total of eight branching fractions of two mass eigenstates decaying into V 1 V 2 . Therefore there are enough experimental measurements to not only solve for the seven unknowns, but also test the hypothesis of mixing between h 0 1 and s. If we observe multiple scalars whose couplings to two electroweak bosons do not follow from that of h 0 1 or h 0 5 , one would be motivated to consider mixing of h 0 1 with an electroweak singlet scalar.
In this section we compute the partial decay width of S → V 1 V ( * ) 2 using the couplings derived in the previous sections. Given that the mass of the scalar could be lighter than the W W threshold, we include the case of S → V 1 V * 2 when one of the vector bosons is offshell. Although decays of an electroweak doublet scalar into two electroweak bosons have been computed both in the on-shell [24] and off-shell [25][26][27] cases, off-shell decays of an electroweak singlet scalar into two electroweak bosons do not appear to have been considered to the best of our knowledge. In the appendix we compute the decay width of a massive spin-0 particle into two off-shell vector bosons, which serve as the basis of the discussion in what follows.
From Eq. (76) in the appendix decays of non-electroweak singlet scalars into W W and ZZ are given by where x = m 2 V /m 2 S , δ W = 2 and δ Z = 1. In the limit x 2 ≪ 1, which is a good approximation if m S is much larger than the ZZ threshold, the pattern of a scalar decaying into two In terms of branching fractions, normalized to the branching ratio into W W , we have where . Custodial symmetry then predicts unique patterns of decay branching fractions for h 0 1 and h 0 5 : We see that a simple counting experiment would allow us to infer the SU(2) C quantum number of the decaying scalar! In Fig. 1 we plot the ratio Br(ZZ/W W ) for an SU(2) C singlet and a 5-plet, including the full kinematic dependence of the gauge boson masses, for the scalar mass between 115 GeV and 1 TeV. We include the decay into off-shell vector bosons using the expression in Eq. (75) for the scalar mass below the W and/or Z threshold. Fig. 1 is the unique prediction of custodial symmetry. Any deviation would imply either the electroweak singlet nature of the scalar or significant violation of custodial symmetry, which in turns suggest cancellation in the ρ parameter at the percent level.
On the other hand, using Eqs. (76), (79), and (80) in the appendix, an electroweak singlet has the following the partial decay widths into two on-shell electroweak bosons where the g sV 1 V 2 couplings are given in Eq. (16). The pattern of partial decay widths into two electroweak bosons is then, again ignoring the effect of gauge boson masses, where V 1 V 2 = {ZZ, Zγ, γγ}, and δ V 1 V 2 is 2 for Zγ and 1 otherwise. This pattern is generically different from that in Eq.  and γγ, normalized to W W mode, could be predicted as follows: In Fig. 2 we plot the predicted Br(Zγ/W W ) and Br(γγ/W W ) branching fractions in terms of Br(ZZ/W W ). Experimental verification of these relations would be a striking confirmation of the singlet nature of the scalar resonance.
By inspection of Eq. (16) we see that a special case occurs when κ 2 = κ 1 , giving Br s (ZZ/W W ) = 1/2, similar to that of h 0 1 . However, in this case we have up to corrections due to the mass of the Z boson. By considering all four partial widths into the electroweak bosons it is still possible to distinguish a singlet scalar from the Higgs doublet even in this special case. However, as commented in the end of Section II, such a scenario lacks any obvious physical motivation.
Another special case is when κ 1 =0, which occurs in the event that the new fermions inducing the dimension-five operators in Eq. (13) carry only hypercharge and no isospin.
This case is not included in Fig. 2 since the partial width of the scalar decaying into W W vanishes! Nevertheless, there would still be significant decay branching fractions into ZZ, Zγ, and γγ states, as predicted by Eq. (16).  In Table I  If one makes the assumption that the individual partial decay width of a scalar decaying into to V 1 V 2 could be obtained, presumably in a future lepton collider or with a very high integrated luminosity at the LHC, then we could explore the possibility of determining the (N L , N R ) multiplet structure under SU(2) L × SU(2) R . The specific question one could ask, given that the SU(2) C singlet from all (N L , N R ) multiplet has the same ratio of couplings to W W and ZZ, is whether it is possible to distinguish the SU(2) C singlet contained in a (2 L , 2 R ) from that contained in a (3 L , 3 R ). To this end we observe that the couplings, g h 0 1 W W and g h 0 1 ZZ in Eqs. (35) and (36), and the gauge boson masses in Eqs. (31) and (32) are given by two parameters: N and the scalar VEV v. Solving for v in terms of the masses and N we obtain Therefore the coupling becomes stronger as N increases. The Higgs doublet has N = 2, while the coupling of the h 0 1 in the (N L , N R ) is (N 2 − 1)/3 times larger than that in the Higgs doublet, resulting in a partial decay width that is (N 2 − 1)/3 enhanced. Once N is known, the complete SU(2) L ×U(1) Y quantum number of the scalar resonance is determined.
As an example, at the LHC one could consider the production of the scalar in the vector boson fusion channels W W/ZZ → S → W W and W W/ZZ → S → ZZ, which provide estimates of The total width Γ t could be extracted by measuring the Breit-Wigner shape of the invariant mass spectrum in the ZZ channel. Then one could simply fit the partial widths Γ W W and Γ ZZ using the different hypothesis for N. Since the event rate in this case is proportional to Γ 2 W W/ZZ , if the total width remains the same the enhancement of a N ≥ 3 multiplet over the Higgs doublet is (N 2 − 1) 2 /9 ≥ 64/9 ≈ 7, which is a significant enhancement.

V. DISCUSSION AND OUTLOOK
We have performed a general analysis up to dimension five of the couplings between electroweak vector boson pairs V 1 V 2 and a Higgs look-alike S, assumed to be a neutral CPeven scalar resonance. We used the framework of unbroken custodial symmetry to group the possibilities into three "pure cases": scalars whose electroweak properties match a SM Higgs, scalars that are SU(2) L × SU(2) R singlets and thus couple to V 1 V 2 only at dimension five, and scalars that couple to V 1 V 2 as a 5-plet under custodial SU(2) C . Fig. 1 shows that it should be straightforward to experimentally distinguish the 5-plet case from the SM-like case of a custodial singlet, using just the ratio of the ZZ and W W decay rates. Fig. 2 illustrates that SU(2) L × SU(2) R singlets produce distinctive relations between the various ratios of V 1 V 2 decay rates, emphasizing the importance of detecting all four decay channels: W W , ZZ, γγ, and Zγ.
To implement our proposal one can either try to extract ratios of partial decay widths directly [28], or measure the individual partial decay widths into pairs of electroweak vector bosons first [29,30] and then take the ratios. In the first possibility the event rate measured in each decay channel of a scalar resonance S is given by Therefore one could approximate the ratio of partial decay widths by the ratio of event rates in each channel, which are measured directly in collider experiments. It would be interesting to study ways to improve on the uncertainty arising from either possibilities.
Since experimental analyses are often driven by final states observed, our study demonstrates the importance of having a correlated understanding of all decay channels into pairs of electroweak vector bosons to avoid misidentification. Tables I and II show how one can be badly fooled by measuring only two of the electroweak V 1 V 2 decay channels for a candidate Higgs. The tables were generated from the predicted properties of a neutral CP -even spin 0 "Higgs" that is in fact an SU(2) L × SU(2) R singlet imposter. In Table 1 the coefficients κ 1 , κ 2 of the dimension-five operators in Eq. (13) have been adjusted so that the ratio of branching fractions of S → ZZ over S → W W coincides with the SM value for the given masses m S . In Table II the same coefficients have been adjusted so that the branching ratio of S → γγ over S → W W coincides with the SM value. In both cases measurement of the two remaining V 1 V 2 decay rates unmasks the Higgs imposter in dramatic fashion.
In a real experiment, the analysis suggested here could be folded into hypothesis testing based on likelihood ratios designed to expose the spin and CP properties of new heavy resonances [4,5]. Higher order effects could be included, as well as the uncertainties associated with unfolding the experimental data to extract the S → V 1 V 2 production and decay properties.
Last but not least we will also need an expression for the two-body phase space: where θ, φ are the polar and azimuthal angles between the direction of V 2 and some other reference direction, e.g. the direction of the boost from the lab frame to the S rest frame, or the direction of the beam. Note that It is important to remember that when V 1 , V 2 are distinguishable particles, we integrate θ, φ over the full 4π solid angle. However when V 1 , V 2 are identical particles (e.g. two Z's or two γ's) we should only integrate θ from zero to π/2, to avoid counting the same final state configuration twice. Thus the angular integration gives 2π in this case, not 4π.
The differential off-shell decay width can be written: where δ V = 1 for identical vector bosons and 2 otherwise. Here Γ µν SV 1 V 2 is the SV 1 V 2 coupling tensor that can be read off from the Lagrangian. The propagator factors become just δ(m 2 i − M 2 V i ) in the narrow width approximation. We will write the coupling tensor as where the coupling constants g h and g s are defined as coefficients of the following operators In the standard modelg 2 hV 1 V 2 = 8m 2 1 m 2 2 G F / √ 2 for W W and ZZ channels and all other couplings vanish at the tree-level, while for an electroweak singlet scalarg hV 1 V 2 = 0. By angular momentum conservation the only nonvanishing contributions from the helicity sums are for (λ 1 , λ 2 ) = (±, ±), or (0, 0): Γ µν ǫ * µ (λ 1 )ǫ * ν (λ 2 ) 2 = |g hV 1 V 2 | 2 (2 + γ 2 a ) + m 2 1 m 2 2 m 2 S |g sV 1 V 2 | 2 (2γ 2 a + 1) where ℜ(c) is the real part of the complex number c. Then the off-shell decay width is The total decay width of S → V * 1 V * 2 is given by The above formula is valid even when the scalar mass crosses the mass thresholds of W and Z bosons. More explicitly, when both vector bosons are on-shell, For a standard model Higgs boson, h, we recover the well-known expression [24] Γ In the case of S → Z * γ, we have to take into account that only the transverse polarizations contribute, and take the limit m 2 → 0. As m 2 → 0 When the Z is on-shell this becomes The width for S → γγ follows from this (note we divide by 2 to get the correct phase space):