Top Spin and CP from Equivariant Transformers

Introduction

Top physics is at the heart of the LHC-precision program, as the top, with its large mass and large production rate, provides a unique experimental probe of fundamental particle properties. Its short lifetime implies that it decays before hadronic bound states can form, so the decay transfers information about the top polarization directly to its decay products [1]. Top-pair spin correlations were first measured in purely leptonic decays. Only recently have they been extended to lepton-hadronic decays [2, 3, 4, 5, 6, 7, 8, 9, 10]. Independent of a potential interpretation in a quantum mechanics framework [11, 12, 13], spin correlations in top pair production conclusively probe $\mathit{CP}$ violation in the top sector [14, 15, 16, 17].

The cleanest spin analyzers in the top decay are the charged lepton and the down-type quark [18, 19]. For hadronic decays, the bottleneck is the identification of the down quark. A naive first approach simply identifies the softer light-quark jet with the down quark. Better results can be achieved with a calibrated down-quark probability [20, 21, 22, 23], but remain far from optimal.

Modern machine learning is transforming every aspect of LHC physics [24, 25], where the state-of-the-art architecture is transformers [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. LHC physics is governed by fundamental symmetries, which can be incorporated explicitly to make transformers more efficient and accurate [37]. Lorentz-equivariant transformers [38] can be constructed using geometric algebra (L-GATr) [33, 36] or boosting between separate reference frames (LLoCa) [35]. We will improve the polarization measurement in hadronic top decays using a state-of-the-art L-GATr-slim classifier [36].

$\mathit{CP}$ violation is introduced into top pair production via dimension-6 SMEFT operators [40, 41]. Specifically, a complex Wilson coefficient of the chromomagnetic dipole operator generates $\mathit{CP}$-odd interference. It can be tracked via $\mathit{CP}$-odd observables, requiring spin information and built via the Levi-Civita tensor [42, 43, 44, 45, 46, 47]. Optimal observables building on this $\mathit{CP}$-odd observable exploit the full phase space correlations. At the detector, these optimal observables are hard to construct explicitly [47], motivating the use of ML-methods in the search for $\mathit{CP}$ violation [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. We will again use a Neyman-Pearson-optimal L-GATr-slim classifier, but now to distinguish $\mathit{CP}$-conjugate WBF-Higgs and top pair events and construct the log-likelihood ratio [61] as an optimal $\mathit{CP}$ observable.

In § 2, we first review the extraction of the top polarization in hadronic decays and the construction of $\mathit{CP}$-odd observables for top pairs. In § 3 we use L-GATr-slim [36] to tag down-type quarks in hadronic top decays, which boosts the spin analyzing power significantly. In § 4 we modify L-GATr-slim to separate scalar and pseudoscalar outputs and demonstrate how this leads to learned $\mathit{CP}$-even and $\mathit{CP}$-odd observables. We validate our result for WBF-Higgs production at parton level and for top pair production including detector effects. We conclude in § 5.

Top polarization and CP-violation

In top decays, the subsequent W-decay determines the final state,

The degree of top spin polarization $P$ is defined through the helicity angle $\theta_i$ between the decay particle $i$ and the top spin axis,

This formula also defines the spin analyzing power $\beta_i$ as the amount of polarization information carried by the decay product $i$. To leading order it is [21]

For the $\bar{t}$ decay, the sign of $\beta_i$ is flipped. Tagging a down quark is challenging, and the associated uncertainties can wash out the polarization information. This points to the easily identified and reconstructed lepton, at the price of significantly lower statistics, especially when correlating decays of top pairs.

Hadronic top decays

We estimate the spin analyzing power from the unit 3-momentum of the down quark $\hat{k}_d$. For hadronic decays with a $b$-tag, the down quark assignment is

ordered by energy in the top rest frame. Following Eq. (2), the top decay products are distributed jointly over phase space in the top rest frame with a spin axis $\hat{n}$ and this down quark assignment,

This corresponds to the optimal $\beta_d=1$. The marginal kinematic distribution is given by the superposition

with

We define the optimal down-probe vector to recover the form of Eq. (5),

We approximate this optimal vector by a unit probe for the down-type 3-momentum $\tilde{k}_d(x)$ and the corresponding helicity angle $\tilde{\theta}_d$. Its kinematic distribution becomes

Here we use the isotropy of the distribution to integrate out the part of $\vec{k}_{\mathrm{opt}}$ orthogonal to $\tilde{k}_d$. The corresponding spin analyzing power is

A simple historical approximation is to identify the down quark with the softer $W$-decay quark

Better probes can be constructed by estimating $p(d=q_{\mathrm{soft/hard}}|x)$. Using the angular distribution of light quarks in the $W$ rest frame with respect to the $b$ momentum gives [21]

Modern ML allows us to estimate the probability from the constituent kinematics as well as their charge and particle IDs. A graph neural network based on ParticleNet [62] yields [23]

In § 3 we will look at ways to improve this down-quark probe using state-of-the-art transformers.

Top pairs and $CP$ violation

While individual tops at the LHC are unpolarized, the spins in (associated) top pair production are correlated,

$B_1$ and $B_2$ are polarization vectors and $C$ is the spin correlation matrix. Experimental measurements mostly use the lepton-lepton decays [2, 3, 4, 5, 6, 7]. The first study of the lepton-hadron channel performed worse [8], but new measurements benefit from more data and ML [9, 10].

$CP$ violation can be introduced into top pair production via a dimension-6 operator with a complex coefficient [63],

This interaction leads to a $CP$-violating interference in the $t\bar{t}$ cross section

It can be probed by $CP$-odd observables of the general form [44, 47]

An additional $CP$-even prefactor can be adapted to the kinematics induced by the $CP$-violating operator. This observable requires four independent 4-momenta, so for a $2\to2$ process we need spin information, for instance, decay 4-momenta. At the parton level

where $p_4$ depends on the decay channel. In the $t\bar{t}$ rest frame, $k_t=(\sqrt{s}/2,\pm\vec{k}_t)$ gives the triple product

It is directly related to the antisymmetric part of the spin correlation matrix $C$ in Eq. (13) [46].

Hadronic top-decay polarization

To illustrate our reconstruction of the top polarization by classifying the two $W$-decay jets, we generate top pair events at the parton level using MadGraph [64] with fixed polarizations

where 0 stands for unpolarized. We simulate lepton-hadron decays with MadSpin, evenly split into hadronic top and anti-top events. We shower the events with Pythia8 [65] and simulate detector effects with Delphes3 [66]. We reconstruct jets with FastJet [67] using the C/A algorithm with $\Delta R = 0.4$ and 1.5.

We require each event to have at least one fat jet, exactly one hard lepton, and exactly two $b$-tagged jets. The $p_T$-leading fat jet is required to have

We de-cluster the fat jet [68, 69] again using FastJet, following Ref. [23]. We revert the clustering sequence until we reach at least three subjets with $m_{\mathrm{subjet}} < 30\ \mathrm{GeV}$. We identify the $b$-subjet and the two light subjets by matching them to the top decay partons within $\Delta R < 0.4$. We modify the mass window of Ref. [23] to [70]

We then reconstruct the leptonic top decay using the $m_W$ criterion to extract $p_{z,\nu}$ and combine the reconstructed $W$ with the second $b$-tagged jet in the event, using the same top mass window.

After reconstruction, we are left with $1.98 \cdot10^{6}$ unpolarized top jets. We use 80%, 10%, and 10% of them for training, validation, and testing. We use $2.41 \cdot10^{5}$ polarized top jets for evaluation, with an equal share of LL, LR, RL, and RR polarizations.

We store the set of 4-momenta of a single top in fixed order:

1. the hadronic top candidate;

2. the $b$-tagged subjet;

3. the light subjets, ordered by their energy in the hadronic top rest frame; and

4. all constituents grouped by the subjet in which they appear, descending in $p_T$.

We ignore all information about the leptonically decaying top in this section. We then add a set of precomputed features per particle that are standard for jet tagging,

Finally, we encode the jet structure, particle ID, and charge. The 4-momenta correspond to vector inputs and the remaining inputs are scalars. We refer to App. A for details.

L-GATr-slim

We employ the Lorentz-equivariant transformer implementation L-GATr-slim [36] to improve the hadronic top polarimetry. For applications under realistic conditions, it is important to train equivariant transformers including a slight symmetry breaking, that we explain in App. B.

Typically, a transformer input token contains all information about one (reconstructed) particle. L-GATr embeds all inputs in the Clifford algebra $\mathrm{Cl}_{1,3}(\mathbb{R})$, as the Lorentz group has a natural representation that commutes with its operators. The 16-dimensional multivector elements of the algebra include a scalar, vector, bivector, axial vector, and pseudoscalar, $(s,v,b,a,p) \in\mathbb{R}^{1} \times\mathbb{R}^{4} \times\mathbb{R}^{6} \times\mathbb{R}^{4} \times\mathbb{R}^{1}$.

An input particle is described by its 4-momentum $v$ and a set of scalar features. The outputs of the comprehensive L-GATr implementation are the elements of the geometric algebra. LHC tasks that only require scalars and vectors motivate the L-GATr-slim implementation: it drops all other multivector elements without any performance loss across regression, classification, and generation [36].

Down-type tagger

Following § 2, an optimal extraction of the top polarization requires tagging the down-type quark, $d=q_{\mathrm{soft/hard}}$. We train the L-GATr-slim classifier on $1.58 \cdot10^{6}$ unpolarized jets and give all hyperparameters in App. § 8. We average the results over five runs and obtain the uncertainty

from the five independent seeds and 50 bootstrap re-samplings of the test dataset.

In the left panel of Figure 1 we see that an identification of the flavor of the $W$-decay jets is possible, with an AUC of $0.785 \pm0.001$. We evaluate the tagger output as the down-type probability of the harder $W$-decay jet. The optimal probe of Eq. (7) requires this output to be a calibrated probability.

Vector (PDF)

In the right panel of Figure 1 we bin the test events in the predicted $p(d=q_{\mathrm{hard}})$ in 20 bins and show the fraction of events in each bin for which the down quark is indeed the harder $W$-decay quark. The curve follows the diagonal over the full range, which means the classifier is correctly calibrated.

Next, we use the classifier output to build the down momentum probe $\tilde{k}_{d}$ and determine the spin analyzing power of our new approach. We define the L-GATr-slim probe as

and reach the currently leading spin-analyzing power

We can use the classifier-based down quark probe to predict the helicity angle and compare it with the parton-level truth. The helicity angle is only meaningful for polarized tops. We pass the left-handed and right-handed polarized tops to the trained classifier and calculate again $\tilde{k}_d$ according to Eq. (25). The helicity angle is then

where we choose the top spin axis $\hat{n}$ as the 3-momentum of the leptonic top in the hadronic top rest frame, although different conventions exist.

In the left panel of Figure 2 we show the distribution of the helicity angle, split into hadronic top polarizations. While the parton-level expectation is a linear trend with a positive (negative) gradient for right-(left-)handed tops, the distributions of $\cos\theta_d$ drop at the edges at the detector level. The distributions using the L-GATr-slim tagger reproduce the peaks around $\pm0.5$, the reduced slope of the linear trend reflecting the imperfect down-tagging.

Vector (PDF)

Finally, we follow Ref. [23] and report the spin analyzing powers at different signal efficiencies. The efficiency is defined as the fraction of events sorted by confidence, i.e. the events where $|p_{\mathrm{L-GATr}}(d = q_{\mathrm{hard}}) - 0.5|$ is the largest, such that a smaller efficiency leads to larger $\beta_{\mathrm{L-GATr}}$. In the right panel of Figure 2 we show the spin analyzing powers, individually extracted for left-handed and right-handed tops, as a continuous function of the efficiency. The L-GATr-slim tagger does not develop a preference for one of the polarization states and tags down-type quarks from left-handed and right-handed tops with the same accuracy at each efficiency. The top 20% events with the highest network confidence achieve an average spin analyzing power of 0.935 $\pm$ 0.002.

CP observable from equivariance

Moving from the polarization of a single top to the spin correlation of top pairs as a probe of $\mathcal{CP}$ violation, we use the same L-GATr-slim classifier to construct an optimal $\mathcal{CP}$-odd observable. In contrast to the original L-GATr, the default version of L-GATr-slim only includes scalars, invariant under $O(1,3)$, and vectors, covariant under $O(1,3)$. To extract $\mathcal{CP}$ information, we need to process $\mathcal{CP}$-even and $\mathcal{CP}$-odd information separately.

CP in L-GATr-slim

Similarly to the original L-GATr, we now include the pseudoscalar multivector element from Eq. (23) in L-GATr-slim. We refer to Ref. [36] for a description of the L-GATr-slim layers and to App. B for more details on the symmetry group and just summarize the extended structure:

• In the linear layer, pseudoscalars do not include a bias which would be $\mathcal{CP}$-invariant,

• The nonlinearity is a modified version of the Gated Linear Unit (GLU) [71] used in L-GATr-slim. We apply non-linear activation functions to Lorentz invariants,

and multiply them element-wise with a linear layer

• The layer norm is modified over the original L-GATr-slim: each multivector channel is normalized separately for numerical stability instead of using a shared norm,

where $c_{x}$ is the number of $x$ channels.

• The attention uses the fact that the Minkowski inner product and the product of two pseudoscalars are both Lorentz-invariant,

with

• To allow the network to separate pseudoscalar information from the vectors, we build $\mathcal{CP}$-odd observables following Eq. (16)

We change the number of vectors from $c_v$ to four times the number of pseudoscalars with a learned projection $w$, and we compute a 4-determinant for each pseudoscalar

Additional operations mixing the three multivector elements are not needed, thanks to the expressivity of the attention.

We stack the layers like in L-GATr-slim, with the PseudoProjection acting on the output of the attention,

This architecture splits the input variables into scalars and pseudoscalars such that the outputs $s(x)$ and $p(x)$ are exactly $\mathcal{CP}$-even and $\mathcal{CP}$-odd. We define the final output as the sum of the two because any real function can be written as the sum of even and odd contributions.

Learned $\mathcal{CP}$ observable

Instead of constructing $\mathcal{CP}$-odd observables by hand, we can use a classifier. Consider two $\mathcal{CP}$-linked probability distributions related by

An optimal classifier between these two phase space densities satisfies

or

in terms of the usual sigmoid required for a probabilistic classifier output. The learned optimal $\mathcal{CP}$-observable is the argument of the classifier-specific sigmoid,

The expanded L-GATr-slim splits any phase space distribution into a scalar and a pseudoscalar contribution,

During training, the learned scalar contribution $s(x)$ will converge to zero and leave us with

WBF Higgs production

We first test our approach on naive WBF Higgs production without decays

where we know the optimal $\mathcal{CP}$-odd observable analytically [47, 52, 59, 72, 73]. We use MadGraph [64] to generate $10^{6}$ parton-level events for three values of the $\mathcal{CP}$-odd SMEFT coefficient

The ordered set of 4-momenta of the Higgs boson and quarks in the final state are the network input. We split the dataset into $1.6\cdot10^{6}$ training, $2\cdot10^{5}$ validation, and $2\cdot10^{5}$ test events. We train five networks per task and quote the mean AUC, with the standard deviation over the trainings added in quadrature to the bootstrap uncertainty, as in Eq. (24).

First, we train L-GATr-slim to separate events with

Because the optimal $\mathcal{CP}$-observable includes a $\mathcal{CP}$-even prefactor, as shown in Eq. (16), this classification requires $\mathcal{CP}$-even and $\mathcal{CP}$-odd information. Its challenging nature is reflected in a moderate AUC of $0.607\pm0.001$.

We can examine the $\mathcal{CP}$ structure of the classifier by looking at the scalar and pseudoscalar outputs in Figure 3. We see that $s(x)$ poorly separates the SM and BSM events in the mean. The scalar separation is driven by $\lvert\mathcal{M}_{\mathcal{CP}}\rvert^{2}/\lvert\mathcal{M}_{\mathrm{SM}}\rvert^{2}$, which varies more for $c_{H\widetilde W}=+1$ than for the SM. For the pseudoscalar output $p(x)$, the SM events without $\mathcal{CP}$ violation remain zero on average, but the events with $\mathcal{CP}$-odd interference give a positive output. This way, L-GATr-slim disentangles the $\mathcal{CP}$ property of the underlying density.

Vector (PDF)

Next, we train L-GATr-slim to differentiate between positive and negative interference,

This classification only uses $\mathcal{CP}$-odd information, so the pseudoscalar L-GATr-slim output is optimal, as discussed in Eq. (40). It is a simpler task than before, and we achieve an AUC of

0.691 $\pm0.002$. In Figure 4 we confirm that the separation relies only on the pseudoscalar output. The scalar output $s(x)$ converges towards zero for positive and negative interference events, indicating that the network learns that the data contains no separable $CP$-even information. In contrast, the pseudoscalar output $p(x)$ splits the dataset into the components with positive and negative interference.

Vector (PDF)

Top pair production

In this main results section, we show how to classify the $CP$ properties of top pair production at the detector level. As discussed in § 2, we induce $CP$ violation through an imaginary part of $c_{tG}$. We generate events using the SMEFTsim top model [74] with [75]

Aside from the SMEFT model, the event generation follows § 3.

We proceed in three steps. First, we use the L-GATr-slim down tagger from § 3 to construct the standard $CP$-odd observable as a benchmark. Second, we identify $CP$ violation by training L-GATr-slim to separate SM events from events with $\operatorname{Im}(c_{tG}) > 0$. Third, we analyze the $CP$ violation by determining the sign of $\operatorname{Im}(c_{tG})$ with the pseudoscalar output as a learned optimal $CP$-odd observable.

$CP$-odd observable from down-tagging

We consider that purely leptonic top pairs are equivalent to our lepton-hadron events with perfect down quark identification and set $\beta_{\ell} \equiv\beta_{d} = 1$. For lepton-hadron pairs, we assign the probabilistic tagging output for the down quark and adapt Eq. (17) to construct a 4-momentum from $\widetilde{k}_{d}$ as a proxy for the true $k_{d}$. The corresponding $CP$-odd observables are

The asymmetries of these $CP$-odd observables separate events with

In Figure 5 we show their distributions for different Wilson coefficients. For lepton-lepton events, the efficiently available polarization information means that differences between the three contributions are more pronounced. On the other hand, the lepton-hadron events benefit from roughly ten times more events due to the branching ratio and the acceptance. To illustrate the power of the $CP$-odd observables, we form the asymmetry

Vector (PDF)

where $N^{\pm}$ is the number of events with $O_1(\operatorname{Im}(c_{tG}))\gtreqless0$. For lepton-lepton and lepton-hadron top decays, the complex Wilson coefficients lead to the asymmetries

The uncertainties here are statistical uncertainties of the dataset and do not correspond to expected measurements.

Identifying $CP$ violation with L-GATr-slim

Next, we use L-GATr-slim to identify $CP$ violation in top pair events. In analogy to the WBF case, we train a classifier to distinguish

This classification requires $CP$-even and $CP$-odd information, so we expect L-GATr-slim to pick up on both. We train on $1.44\cdot10^{6}$ events and evaluate on $1.80\cdot10^{5}$ events. We provide further details on the network inputs and hyperparameters in App. § 8.

In Figure 6 we show the scalar and pseudoscalar classification outputs. In combination, L-GATr-slim reaches an AUC of $0.565\pm0.002$ with the true down quark and $0.556\pm0.002$ with the tagged down quark. We observe the same behavior as in the WBF case despite the lower performance. The scalar output separates the SM and BSM events in the mean and is broader for $\operatorname{Im}(c_{tG}) = 2.6$, where the $\mathit{CP}$-even term $|\mathcal{M}_{\mathit{CP}}|^2$ modifies the kinematics. The pseudoscalar output averages to zero for the SM events and to a positive value for $\operatorname{Im}(c_{tG}) = 2.6$, which includes the $\mathit{CP}$-odd interference. Its shift is much smaller than the scalar one, so the $\mathit{CP}$-even information leads this classification.

Vector (PDF)

$\mathit{CP}$-analysis from L-GATr-slim

Finally, we show how we can analyze $\mathit{CP}$ violation in top pair events using the $\mathit{CP}$-odd observable learned by L-GATr-slim. We want to determine the sign of the $\mathit{CP}$-odd interference term

We use the same training setup as for the previous analysis, with $1.56 \cdot10^{6}$ training events and $1.96 \cdot10^{5}$ events for evaluation.

In Figure 7 we show the scalar and pseudoscalar L-GATr-slim outputs. We first see that the scalar output $s(x)$ does not include any information and is moved to zero, just like in § 4.3. The pseudoscalar output $p(x)$ captures the $CP$-odd information and shows a clear separation of the two signs early in the training. We reach an AUC of $0.551 \pm0.002$.

Vector (PDF)

From Eq. (40) we know that $p(x)$ is the optimal $CP$-odd observable $O_{\mathrm{L\text{-}GATr}}$. As mentioned above, we use the true down quark assignment for lepton-lepton top decays. For lepton-hadron decays, we define the $CP$-odd observable as weighted with the two down quark assignments,

In Figure 8 we show the observable for both decay channels. We also compute the asymmetries

Vector (PDF)

We pick the networks with the best validation loss out of five runs and only include the statistical uncertainty, as computed for $O_{1}$. We compute the asymmetries for $\operatorname{Im}(c_{tG}) = 0.1$ on the full dataset, because the network is not trained on it, whereas for $\operatorname{Im}(c_{tG}) = 2.6$ we only use the test data, hence the larger statistical error. We find that for $\operatorname{Im}(c_{tG}) = 2.6$ the learned observables from L-GATr-slim substantially improve over their $O_{1}$ counterparts in Eq. (49), while they agree within the statistical uncertainties for $\operatorname{Im}(c_{tG}) = 0.1$.

Outlook

Measuring the polarization of tops and correlating the polarizations of top pairs provide us with a powerful probe of BSM physics. In this paper, we have shown how a Lorentz-equivariant transformer significantly improves top polarimetry and searches for $CP$ violation in top pair production. To this end, we have equipped L-GATr-slim with a pseudoscalar channel. This architecture splits scalar and pseudoscalar information and allows us to learn optimal $CP$-odd observables. First, we have reconstructed the polarization of hadronically decaying tops by tagging the down-type quark with a calibrated probability output. The resulting spin-analyzing power of 0.721 at detector level significantly outperforms all existing methods and can be increased by using the most suitable 20% top decays.

We have applied L-GATr-slim to the toy problem of extracting $\mathit{CP}$ violation from parton-level WBF events. Distinguishing SM events from events with dimension-6 $\mathit{CP}$ violation requires scalar and pseudoscalar outputs, the pseudoscalar indicating that the signal indeed violates $\mathit{CP}$. Determining the sign of the $\mathit{CP}$-violating Wilson coefficient only involves the new pseudoscalar L-GATr output.

Finally, we have established our new L-GATr-based methods for $\mathit{CP}$ violation from a complex dimension-6 Wilson coefficient in top pair production, including detector simulation. First, we used the down-type tagger to construct a $\mathit{CP}$-odd observable. Alternatively, we employed the scalar and pseudoscalar multivector elements to learn an optimal $\mathit{CP}$-sensitive observable to extract the effect of the complex dimension-6 operator. Finally, we have shown how to determine the sign of this Wilson coefficient optimally, just using the pseudoscalar L-GATr-slim output.

Our applications show that equivariant transformers based on a geometric algebra give us access to relevant information for BSM physics searches at the LHC through output channels with well-defined Lorentz representations.

Code availability

The code for this paper is available as part of the public Heidelberg HEP-ML code and tutorial library. The L-GATr-slim architecture and its pseudoscalar extension are part of the L-GATr repository. The datasets are available upon request.

Acknowledgements

We thank Jonas Spinner for helpful discussions about the extended L-GATr-slim architecture. This project has received funding from the European Union’s Horizon Europe research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101168829, Challenging AI with Challenges from Physics: How to solve fundamental problems in Physics by AI and vice versa (AIPHY). MM acknowledges funding from the German Research Foundation (DFG) through project number 580386328 and through Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers – 390837967. TP is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762 – TRR 257 Particle Physics Phenomenology after the Higgs Discovery. Computations were performed on the LUIS computing cluster at Leibniz University Hannover funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 563020064, INST 187/742-1 FUGG, and INST 187/592-1 FUGG. The authors would like to thank the state of Baden-Württemberg for its support of bwHPC and the German Research Foundation (DFG) for funding under "Project number 455622343" (bwForCluster NEMO 2).

Network inputs and hyperparameters

Each token in the sequence given as input to the network contains vector, scalar and pseudoscalar information. The different inputs are shown in Table 1. The structure of the jet is given by a one-hot encoding of the fat jets and subjets, passed as scalar inputs. The particle ID is also given as the standard 6-dimensional one-hot encoding. The additional kinematic features passed as scalars and pseudoscalars are computed in the lab frame. We refer to App. § 9 for more details on the choice of scalars and pseudoscalars. We also include the symmetry-breaking vectors from App. § 9 as additional tokens, with their scalars and pseudoscalars set to 0.

		top jet	$b$ subjet	$q_{\mathrm{hard}}$	$q_{\mathrm{soft}}$	$\cdots$	constituent	$\cdots$	leptonic top	lepton
	hadronic top	1	0	0	0		0		0	0
	subjet	0	1	1	1		0		0	0
	in $b$ subjet	0	1	0	0		0		0	0
	in $q_{\mathrm{hard}}$ subjet	0	0	1	0	$\cdots$	1	$\cdots$	0	0
	in $q_{\mathrm{soft}}$ subjet	0	0	0	1		0		0	0
	leptonic top	0	0	0	0		0		1	0
	lepton	0	0	0	0		0		0	1
$s$	electron	0	0	0	0		0		0	0
$s$	muon	0	0	0	0		0		0	0
$s$	photon	0	0	0	0	$\cdots$	0	$\cdots$	0	0
$s$	charged hadron	0	0	0	0		1		0	0
$s$	neutral hadron	0	0	0	0		0		0	0
	$\log p_T$						$\log p_{T,i}$
	$\log E$						$\log E_i$
	$\Delta\log p_T$						$\log p_{T,i}-\log p_{T,J}$
	$\Delta\log E$						$\log E_i-\log E_J$
	$|\Delta\phi|$						$|\phi_i-\phi_J|$
	$|\Delta\eta|$						$|\eta_i-\eta_J|$
	$\Delta R$						$\sqrt{\Delta\phi^2+\Delta\eta^2}$
$v$	4-momentum						$E_i$
$v$	4-momentum						$p_{x,i}$
$v$	4-momentum						$p_{y,i}$
$v$	4-momentum						$p_{z,i}$
$p$	charge	0	0	0	0	$\cdots$	$+1$	$\cdots$	0	$Q_\ell$
$p$	$\Delta\phi\cdot\Delta\eta$						$(\phi_i-\phi_J)(\eta_i-\eta_J)$

The light subjets are ordered by their energy in the hadronic top rest frame. For the $t\bar{t}$ classification, we swap them such that the (predicted or true) down-type subjet is the first one, and the network can identify it with the one-hot encoding. For the down-tagging task, the leptonic top and the lepton are not included, and the $\mathcal{CP}$ symmetry is not needed, so the pseudoscalars are passed as scalars.

All networks use the same backbone and training setup, listed in Table 2. Only the pseudoscalar channels and the number of training iterations differ between the tasks. We use the default hyperparameters of [38] for tagging with L-GATr-slim and did not perform a dedicated optimization.

	Down Tagging	WBF	$t\bar{t}$
Blocks, heads		12, 8
MLP ratio		4
Channels $(s,v,p)$	(96, 32, 0)	(96, 32, 32)	(96, 32, 32)
Optimizer		Lion
Learning rate		$3 \cdot10^{-5}$
LR scheduler		Cosine annealing
Weight decay		2
Batch size		128
Iterations	$5 \cdot10^{5}$	$5 \cdot10^{4}$	$2 \cdot10^{5}$

L-GATr-slim symmetry group

The symmetry group of L-GATr-slim depends on the type of inputs that are given. The network is equivariant with respect to $O(1,3)$, under the assumption that the scalar inputs are invariant and the vector inputs are covariant. If the inputs do not satisfy that property, the network is no longer equivariant: the symmetries that are broken at the input level can also be broken at the output level.

We know that detector data does not respect the full Lorentz group symmetry, so we include in the input three reference vectors that are identical in every event to break those symmetries: $t=(1,0,0,0)$ and $n_{\pm}=(1,0,0,\pm1)$, where $n_{\pm}$ defines the beam axis in the $z$ direction. Because those vectors are fixed and do not transform, they are only covariant under the following transformations:

• rotations around the beam axis $R_z(\theta)$

• rotations by $\pi$ around any transverse axis, for example the beam swap $R_x(\pi)$

• mirrors through planes containing the beam axis, for example $\sigma_y=\mathcal{P}R_y(\pi)$

• rotations around the beam axis combined with the flip $\sigma_z=\mathcal{P}R_z(\pi)$

Therefore, the L-GATr-slim network with those symmetry-breaking vector inputs is only equivariant under

where the rotation $R_x(\pi)$ can be replaced with a rotation around any other axis perpendicular to $z$.

However, we are interested here in the $\mathcal{CP}$ symmetry instead of $\mathcal{P}$. The $C$ transformation only acts on the charges and not on the Minkowski space $\mathbb{R}^{1,3}$, hence it commutes with all elements of $O(1,3)$. Therefore, we can write the remaining symmetry group as

The same issue arises with the scalar inputs: the usual scalars given to a jet tagger, as those in Eq. (22), are only invariant under $SO(2)_z$ and not under $O(1,3)$. Therefore, the standard L-GATr-slim tagger is $SO(2)_z$-equivariant, which was sufficient for previous applications.

In our case, we have to separate the scalars into four categories based on their transformations under $R_x(\pi)$ and $\mathcal{CP}$. The transformations of the relevant inputs are shown in Table 3. We build our architecture to handle the $\mathcal{CP}$ symmetry and neglect the $R_x(\pi)$ symmetry. Therefore, we only include scalars that are invariant under $R_x(\pi)$. We find that it makes no difference in our results to break this symmetry. It could also be built into the network by adding additional channels in the same way we added the pseudoscalar channels, but that presents no benefits here.

$R_x(\pi)$	$CP$	Example	L-GATr-slim input
even	even	$|\Delta\phi|$, $|\Delta\eta|$, $\Delta R$	$s$
even	odd	$(\Delta\phi\cdot\Delta\eta)$, charge	$p$
odd	even	$\Delta\phi$	/
odd	odd	$\Delta\eta$	/

References