Limits on Electroweak Instanton-Induced Processes with Multiple Boson Production

Recently, the CMS collaboration has reported their search for electroweak instanton-like processes with anomalous $B+L$ violation assuming multi-fermion but zero-boson final states. On the other hand, many theoretical studies suggest that anomalous $B+L$ processes may have an observably large production rate only if their final state contains a large number of electroweak gauge bosons. In this paper, we compare collider signatures of zero- and multi-boson events of anomalous $B+L$ violation at the LHC and derive an upper limit on the cross-section for the multi-boson process by recasting the CMS analysis.


Introduction
The discovery of the Higgs boson at the LHC has marked the completion of the Standard Model (SM) of electroweak and strong interactions. Moreover, so far, no compelling evidence for physics beyond the SM has been observed at the LHC. In fact, the perturbative SM predictions for hard, short distance dominated scattering processes are all verified experimentally to a remarkable precision.
Theory predicts, however, that there are also hard scattering processes in the SM whose amplitudes are fundamentally non-perturbative. Their existence follows from axial anomalies [1][2][3] in the SM, resulting in an anomalous violation of baryon plus lepton number, B + L, in electroweak interactions [4,5]. They are induced by topological fluctuations of the non-Abelian SU(2) L gauge fields, notably electroweak instantons [6,7]. Anomalous B + L violating processes are believed to be very important in the high temperature primordial plasma after the big bang [8][9][10] and to have therefore a crucial impact on the evolution of the baryon and lepton asymmetries of the universe (see Ref. [11] for a review).
A very interesting, albeit unsolved question is whether manifestations of such topological fluctuations might be directly observable in high-energy scattering at present or future colliders. This question has been raised originally in the late eighties in [12,13], but, despite a lot of effort, the actual size of the cross-sections in the relevant, tens of TeV energy regime could not be determined.
Nevertheless, it has been established (for reviews, see Refs. [11,14,15]) that the inclusive cross-section for electroweak (B + L)-violating processes, can be written in the form is the quark-quark centre-of-mass energy √ŝ in terms of the scale which is parametrically similar to the energy of the sphaleron, E sph = f (m 2 h /m 2 W )πm W /α W ∼ 9 TeV -the static energy of a classically unstable saddle-point solution of the static bosonic field equations of the electroweak theory, corresponding to the minimal barrier height between inequivalent vacua with different values of the Chern-Simons topological index N CS [16,17]. The 'holy grail function' F in the exponent is known in terms of an expansion in , whose first few terms are given by [18][19][20][21][22] where d 0.15. Obviously, in the energy region m W √ŝ M 0 , the inclusive cross-section for (B + L)violation is unobservably small, due to the tremendous exponential suppression factor, exp(−4π/α W ) ∼ 10 −162 , typical for a quantum tunneling process. Nevertheless, it is exponentially growing in this energy region. Intriguingly, the characteristic scale for the exponential growth is M 0 and thus of the order of the sphaleron energy. One may interpret this as a hint that the process proceeds via an intermediate virtual sphaleron-like field configuration. This interpretation is also backed-up by the fact that the inclusive cross-section (1.2) is dominated by a semi-classically large number of W (Z) and Higgs bosons [24], and 8) in line with the expectation from the decay of a sphaleron-like intermediate state [25]. Unfortunately, however, the perturbative expansion of the holy-grail function F in powers of breaks down around ∼ 1 and therefore it is not possible to infer the size of the cross-section at centre-of-mass energies relevant for the LHC. The multi-boson events such as Eq. (1.1) lead to spectacular collider signatures. According to Eq. (1.7), one expects O(30) electroweak bosons in an event on average. 2 To study collider signatures of Eq. (1.1), the event generator HERBVI [26] was developed, and the first thorough phenomenological study was carried out in Ref. [27].
After the end of the SSC project in the early nineties, the motivation for theorists to decide this question immediately dropped considerably and the research in this direction essentially stopped. 3 However, a recent paper by Tye and Wong [34], presenting qualitative arguments in favour of observable rates for electroweak (B + L)-violation, revived the interest in it and motivated phenomenological [35][36][37][38][39][40] and experimental studies. In particular, the CMS collaboration has recently reported their results of searching for signatures of electroweak instanton-induced processes at the LHC [41].
There is, however, a remarkable difference between the processes considered in the CMS analysis and the virtual sphaleron-like processes discussed in the literature and briefly summarised above. Following earlier studies [35] and for simplicity, CMS postulated the zeroboson process qq → n q q + 3 (1.9) where n q = 7, 9 or 11, and q/ represent quark/lepton or anti-quark/anti-lepton depending on how the fermion lines are contracted with an effective vertex in the instanton background with topological charge ∆N CS = ±1. In the CMS analysis, the event generator BaryoGEN [42] was used to simulate the process in Eq. (1.9).
In this paper we investigate the difference in collider signatures between multi-and zeroboson process (i.e. Eq. (1.1) and Eq. (1.9)) using the HERBVI and BaryoGEN event generators, respectively. After comparing various distributions in section 3, we recast the original CMS analysis and derive the cross-section upper limit on the sphaleron-like multi-boson process in section 4. Section 5 is devoted to conclusions.

Event simulation
It has been suggested that the steeply-rising event rate of sphaleron-like processes can be modelled by a sharp threshold behaviour as a function of the partonic centre-of-mass energy, √ŝ [43]. We therefore postulate the following partonic cross-section: where Θ(x) is the Heaviside theta function; Θ(x) = 1 for x ≥ 0 and 0 otherwise, and E sph is the partonic threshold energy. We take E sph = 9 TeV as the default value for our simulation (later varying it by ±1 TeV), motivated by the fact that the sphaleron barrier height with m h = 125 GeV is given by ∼ 9.1 TeV [16]. In Eq. (2.1), the cross-section is normalised by 1/m 2 W up to the dimensionless parameter p sph , which controls the overall cross-section. The same parametrisation is used in the CMS analysis [41].
The event samples for zero-and multi-boson final states are generated by BaryoGEN [42] and HERBVI [26], respectively, for pp collisions at centre-of-mass energy √ s = 13 TeV. Since BaryoGEN only generates partonic final states from the hard process, the sample is passed to Pythia 8 [44] to simulate the parton shower and hadronization. HERBVI is a hard process generator interfaced to the Fortran HERWIG generator [45]. For this study we use the last Fortran version HERWIG 6.521 [46].
The above hadronic samples are then passed to Delphes [47] to simulate the detector response. We tune the parameters in Delphes in such a way that it emulates the CMS analysis [41] as accurately as possible. Jets are reconstructed using the anti-k T algorithm with cone size 0.4. Leptons (electrons and muons) and photons are required to be isolated from neighbouring energy activity within ∆R ≡ ∆φ 2 + ∆η 2 < 0.4 (muons) or 0.3 (electrons and photons). We require the scalar sum of the transverse momenta p T of the particles within this cone (excluding the targeted object itself (lepton or photon)) to be less than 15% (muons), 10% (electrons) or 12% (photons) of the p T of the targeted object. We demand p T > 30 GeV for all objects (jets, leptons and photons) and |η| < 5 (jets), < 2.4 (muons) and < 2.5 (electrons and photons).

Zero vs high boson multiplicity events
We begin this section by showing the boson multiplicity distributions obtained by HERBVI in Fig. 1. The distributions are obtained as follows. First, the gauge boson multiplicity distribution has been calculated in the symmetric phase using the leading-order matrix element formula given in [12,13,27]. The distributions of Z and γ are obtained by transforming the neutral W bosons into Z and γ with probabilities cos 2 θ W and sin 2 θ W , respectively. One can see in the figure that the W boson multiplicity peaks around 20. One the other hand, the central value of the Z multiplicity is about a factor of 2 smaller. The number of photons per event is rather modest but 2 or 3 photons are expected on average. We do not show the Higgs boson multiplicity distribution because it is negligible compared to that of gauge bosons.
We now compare various distributions obtained for the events with zero-(BaryoGEN) and multi-(HERBVI) boson final states in Fig. 2. In this and the rest of the figures, the bluedashed and red-solid histograms correspond to the distributions for the zero-and multi-boson final state events, respectively. The top left panel shows the jet multiplicity distributions. As expected, the distribution for zero-boson final states peaks around N j = 10, since 7, 9 or 11 quarks are present in the final states coming from the hard interaction, although some extra jets can arise from initialand final-state QCD radiation. On the other hand, N j peaks around 22 in the multi-boson case. The extra jets are coming mainly from the decay of heavy electroweak bosons. It is interesting that 99% of the multi-boson events have N j ≥ 15. In such a high jet multiplicity region the Standard Model background is extremely low.
The lepton multiplicities are compared in the top right panel of Fig. 2. As can be seen, the number of leptons in an event is much larger for the multi-boson process, since the extra leptons are produced in the decay of the heavy bosons. While only ∼ 20% of the zero-boson events have N > 1, this rate is ∼ 40% for the multi-boson events.  A large difference is also found in the photon multiplicity distributions, shown in the bottom panel of Fig. 2. For the zero-boson events, photons are produced only from QED radiation or decays of hadrons, while for the multi-boson process, high-p T photons may be produced also from the primary hard interaction. For the zero-boson case, ∼ 85% of events do not have any such photons and the probability of having more than one photon is extremely small (< 1%). On the other hand, the photon multiplicity distribution peaks around N γ = 1 for the multi-boson case and has a tail towards the higher multiplicity region, N γ = 2, 3. For example, the rate of having three high-p T photons is as large as ∼ 10%.

Limits on sphaleron process
The CMS collaboration has recently carried out an analysis searching for mini black holes and sphalerons that could be produced by the 13 TeV proton-proton collisions at the LHC [41]. The dataset used was collected in 2016 and corresponds to an integrated luminosity of 35.9 fb −1 . The events are recorded if they pass the trigger requirement of H T > 800 or 900 GeV, depending on the period when the data was collected, where H T is defined as the scalar sum of the p T of all jets reconstructed at the stage of the High Level Trigger.
The CMS analysis adopts a simple cut-and-count method. The cuts are imposed on the two variables, N (p T > 70) and S T , where N (p T > 70) is the number of all reconstructed objects (jets, leptons and photons) with p T > 70 GeV and S T is the sum of the transverse missing energy, E miss The following ranges were used for signal regions; N min = 3, ..., 11 and 3.8 < S min T /TeV < 8. CMS performed a careful estimation of the Standard Model background. The dominant background comes from QCD multi-jets, followed by V + jets, γ + jets and tt production. By comparing the number of observed events with the expected background contribution in each signal region, CMS estimated a model-independent 95% CL upper limit on the extra contribution to the signal region. We refer to this bound as N max(a) obs , where a labels the signal region in question. CMS also estimated the "expected" upper limit, N max(a) exp , by assuming that exactly the same (fractional) number of events predicted by the background was observed.
Our procedure for setting limits is as follows. First, we estimate the contribution from sphaleron events to the signal region a, N . We then compare the signal contribution with the observed upper limit only at the most sensitive signal region. We exclude the signal hypothesis if N (a * ) sph /N max(a * ) obs > 1.
In Table 1, we show the most sensitive signal region (a * = (N min , S min T [TeV]) * ), the signal efficiency ( (a * ) ) at signal region a * , and the observed upper limit for signal region a * (N max(a * ) obs ) for different values of the partonic threshold energy E sph . The signal efficiency is defined as the fraction of events that satisfy the condition employed by the signal region. In particular, it satisfies the relation N (a) sph = σ · (a * ) ·L int , where L int is the integrated luminosity and σ is the inclusive hadronic cross-section obtained by convoluting the partonic crosssection in Eq. (2.1) with the parton distribution functions (PDFs). in Fig. 3 we show the inclusive hadronic cross-section with p sph = 1 in Eq. (2.1), where the CT10 LO PDF set [48] was used following the CMS analysis [41].
In Table 1, we see that N min = 11, S min T = 4.2 TeV is selected for the most sensitive signal region throughout the range of E sph for the multi-boson case, whilst N min = 8, 9 and S min T ∈ [5.4, 6.2] are chosen for the most sensitive signal regions depending on E sph for the zero-boson case. The tendency is clear that the multi-boson process favours the high multiplicity and small S T regions, while the zero-boson process prefers the low multiplicity and large S T regions in comparison.
To understand this, we first look at the N (p T > 70) distributions shown in the left panel of Fig. 4. One can see that N (p T > 70) for the zero-boson case peaks around 8, which is significantly lower than the typical multiplicity of the primary partons, N 0 ∼ 12, produced from the hard interaction. In fact, only half of the events have N (p T > 70) ≥ 8. The reason for this may be understood as follows. Since the primary partons from the zero-boson process are only quarks and leptons, N 0 ∼ 12 should be interpreted as the maximum of the N (p T > 70) distribution rather than the average. In fact, among N 0 partons at most three of them may be neutrinos, which will of course not be counted in N (p T > 70). Furthermore, the p T threshold of 70 GeV may have a large impact on the lower p T objects. To see this we show the p T distributions of the 10th hardest object in the event in the right panel of Fig. 4. We see from this plot that the distribution for the zero-boson case peaks near the lowest p T bin, and only ∼ 40% of the 10th hardest objects have p T > 70 GeV. This is not surprising because large differences in p T are expected between the hardest and softest primary objects. For the multi-boson case, this effect is much smaller because the event typically contains many more than 10 primary objects, and the difference in p T between the hardest and 10th hardest objects is milder.
As seen in the right panel of Fig. 4, the lower tail of N (p T > 70) barely reaches 11 for the multi-boson case. This means the the multi-boson signal need not pay any price for the condition N (p T > 70) ≥ N min , since CMS varies N min only up to 11 in their signal regions. In fact, the multi-boson signal efficiencies are always 95% in the most sensitive signal region (N min , S min T ) * = (11, 4.2 TeV). The larger the N min , the lower the Standard Model background, and N min = 11 is therefore selected throughout E sph in the most sensitive signal region for the multi-boson case.
We now look at the S T distributions in Fig. 5 with N min = 11. We see that the values of S T are generally quite high. The distribution peaks around 6 TeV for the multi-boson process and 7 TeV for the zero-boson one. This is expected because the S T variable is designed to reflect the total energy of the event, E sph ∼ 9 TeV. One can also see that S T is generally lower in the multi-boson case compared to the zero boson case. This is because more objects are rejected by the p T > 70 GeV cut. For the multi-boson process, the number of primary partons (including those produced from heavy boson decays) may be estimated as N 0 ∼ O(70) assuming n B ∼ O(30) and each boson decays into two partons. Therefore, the total event energy is distributed among N 0 partons, yielding a large chance for each one to be rejected by the p T > 70 GeV cut. However, the impact of this on S T is not very large. More than 98% of the events are accepted by S T > 4 TeV for both zero-and multi-boson samples.
The left panel of Fig. 6 shows the 95% CL exclusion limits in the (E sph vs σ) plane. The limit, σ < σ max , is obtained from σ max · (a * ) · L int = N max(a * ) obs with L int = 35.9 fb −1 . One can see that a stronger limit is placed on the multi-boson signal than on the zero-boson, although CMS targets the latter in their analysis. As discussed above, this is because the multi-boson processes give rise to larger multiplicity events, enabling one to go to a large N min region, where the Standard Model background is extremely small, without losing signal. The crosssection limit is given by σ max 0.8 fb −1 throughout the range of E sph in the plot for the multi-boson case. This can be seen also in Table 1, since the signal efficiency, (a * ) , and the observed upper limit, N max(a * ) obs , are almost independent of E sph for this case. Since (a * ) and N max(a * ) obs are close to their highest and lowest values, respectively, there may be no room to further improve the limit for the multi-boson case, except by increasing the integrated luminosity or the collider energy.
In the case of the zero-boson final state, the cross-section limit is stronger for larger E sph , since the larger the E sph , the higher the S T on average. On the other hand, the hadronic cross-section decreases rapidly upon increasing the partonic threshold energy, due to the PDF suppression for fixed p sph . We show in the right panel of Fig. 6 the 95% CL excluded regions in the (E sph vs p sph ) plane. As expected the limit on p sph is weaker for larger E sph . We superimpose in this plot the limit shown in the CMS analysis [41]. This should be compared to the zero-boson limit obtained from our recasting of the analysis based on the fast detector simulation with Delphes. One can see that our limit is on the conservative side and reasonably close to the CMS limit.

Summary and conclusions
In this paper we compared various kinematical and multiplicity distributions between the anomalous B + L violating processes with zero-and multi-boson final states. The former final state was assumed in the recent CMS analysis [41] on searching for instanton-induced processes at the LHC, whilst many theoretical studies suggest that the latter is more realistic.
We showed that there is a significant difference in jet multiplicity distributions; the peak positions are around 10 for zero-boson and 22 for multi-boson final states, for jets with p T > 30 GeV. We also found that a multi-boson event contains at least one lepton and photon with more than 50 % probability. On the other hand, the S T variable tends to be smaller for the multi-boson final state because a large number of jets are rejected by the somewhat harsh p T cut (p T > 70 GeV) employed by the CMS analysis.
By applying the same event selection used in the CMS analysis [41], we have derived a cross-section upper limit, σ < 0.8 fb −1 , on the multi-boson processes within the region we considered, 8 ≤ E sph /TeV ≤ 10. We found that unlike the zero-boson case, the most sensitive signal region is identified as the one with the highest jet multiplicity bins, in which the SM background is extremely low and less than 1 event for the current integrated luminosity, 36.1 fb −1 . Therefore, this signal region almost maximizes the analysis efficiency in terms of S/ √ B and further improvement cannot be expected within the current dataset. On the other hand, by employing even larger jet multiplicity bins, e.g. N j ≥ 15, one can expect some improvement on the analysis searching for the sphaleron-like process with multi-boson final states, for different energy or luminosity options.