Factorization for substructures of boosted Higgs jets

We present a perturbative QCD factorization formula for substructures of an energetic Higgs jet, taking the energy profile resulting from the $H\to b\bar b$ decay as an example. The formula is written as a convolution of a hard Higgs decay kernel with two $b$-quark jet functions and a soft function that links the colors of the two $b$ quarks. We derive an analytical expression to approximate the energy profile within a boosted Higgs jet, which significantly differs from those of ordinary QCD jets. This formalism also extends to boosted $W$ and $Z$ bosons in their hadronic decay modes, allowing an easy and efficient discrimination of fat jets produced from different processes.

devoted to the exploration of heavy particle jet properties based on event generators [3][4][5]. Recently, the perturbative QCD (pQCD) formalism, including fixed-order calculations [6] and the resummation technique [7], was employed to investigate jet substructures. The alternative approach based on the soft-collinear effective theory and its application to jet production at an electron-positron collider were presented in Refs. [8,9].
In this Letter we develop a pQCD factorization formula to describe the internal jet energy profile (JEP) of the boosted jet resulting from the H → bb decay, with energy E JH and invariant mass m JH . The basic idea of our theoretical approach is as follows. A Higgs boson is a colorless particle, while its decay products, the bottom quark and antiquark, are colored objects and dressed by multiple gluon radiations to form a system with mass of O(m JH ) and energy of O(E JH ). The invariant mass m J of the bottom quark and its collimated gluons, with energy of O(E JH ), typically satisfies the hierarchy E JH ≫ m JH ≫ m J . Based on the factorization theorem, QCD dynamics characterized by different scales must factorize into soft, collinear, and hard pieces, separately. First, the Higgs jet function J H is factorized from a Higgs boson production process at the leading power of m JH /E JH . Then the b-quark jet function is defined at the leading power of m J /m JH [7], soft gluons with energy of O(m JH ) are absorbed into a soft function S, and the remaining energetic gluons with energy O(E JH ) and invariant mass of O(m JH ) go into the hard Higgs decay kernel H.
The Higgs JEP is then factorized at leading power of m J /m JH into a convolution of the hard kernel with two b-quark jet functions and a soft function that links colors of the two b quarks. We will demonstrate a simple scheme, in which the soft gluons are absorbed into one of the b-quark jets, forming a fat jet, and the soft function reduces to unity. The other b-quark jet is a thin jet to avoid double counting of soft radiation. Evaluating the decay H → bb up to leading order (LO) in the coupling constant α s and substituting the light-quark jet function in [7] for the b-quark jet functions, we predict the Higgs JEP. Since a Higgs boson is massive and a color singlet, its JEP dramatically differs from that of ordinary QCD jets. Below, we present the derivation of the JEP of a Higgs boson decaying into a bottom-quark pair.
The four-momentum of the Higgs jet can be written as P JH = E JH (1, β JH , 0, 0), with β JH = 1 − (m JH /E JH ) 2 . We define the Higgs jet function at the scale µ as where the coefficient has been chosen to satisfy J (0) H = δ(m 2 JH − m 2 H ) at the zeroth order in the Yukawa coupling. m H represents the Higgs boson mass, and R the Higgs jet cone radius. The three δ-functions in the above definition specify the Higgs jet invariant mass, energy, and unit momentum direction of the set N J of final-state particles, respectively. After applying the aforementioned factorization procedure, J H is written as where the factorization scale µ f is introduced by the b-quark jet functions J i . m Ji (E Ji , P Ji , R i ) is the invariant mass (energy, momentum, radius) of the b-quark jets, and the soft function takes the form S (0) = δ(ω) at LO with the variable ω ≡ P S · P JH , where P S is the soft gluon momentum.
To describe the Higgs JEP, we define the jet energy function J E H (m 2 JH , E JH , R, r, µ 2 ) by including in Eq. (2) a step function Θ(r − θ j ) for every final-state particle j. The final-state particles with non-vanishing step functions (i.e., emitted within the test cone of radius r) and associated with the b-quark jet J i are grouped into the b-quark jet energy function The energetic final-state particles outside the b-quark jets and within the test cone are absorbed into the hard kernel H E . The other finalstate particles outside the test cone are absorbed back into their original functions. In this work, we will consider only the LO hard kernel, for which H E = H (0) . We then arrive at where the LO hard kernel is with the number of colors N c , the b-quark mass m b , the vacuum expectation value v, the Higgs decay width Γ H , and the polar angle θ Ji of the b-quark jet J i relative to the Higgs jet axis.
We have the freedom to choose the jet parameters R i and r i , whose values depend on the scheme adopted to factorize the soft radiation in the Higgs jet into different convolution pieces in Eq. (3). A simple scheme is to take J 1 as a thin jet, such that its entire energy is counted when a sufficient amount of the thin jet is within the test cone as specified below. For that, we set R 1 = r 1 = r, which increases from the minimal value 0.1 in our numerical analysis. This choice leads to the simplification of J being a geometric factor. The scheme also includes a fat jet J 2 with a large cone radius R 2 = R, which then absorbs all soft radiation in the Higgs jet. The energy function of the fat jet J E 2 (m 2 J2 , E J2 , R, r 2 = r, µ 2 f ), which contributes to the Higgs JEP as θ J2 ≤ ar, will take the result derived in the resummation technique [7,10].
Next, we integrate out the dependence on the Higgs jet invariant mass by taking the first moment in the To perform the integration overn J2 , we write the corresponding δ-function as where the ratio is given by |P where the color factor C F = 4/3 and the momentN ≡ N exp(γ E ), with γ E being the Euler constant. The off-shellnessξ 2 Ji associated with the b-quark jets implies that S (1) contains the collinear dynamics which has been absorbed into the jet functions. Hence, the subtraction of the collinear divergences from the soft function is necessary to avoid double counting. The collinear divergences from loop momenta collimated to the b quark (b quark) can be collected with theb quark (b quark) line being replaced by the eikonal line in the direction n J1 (n J2 ) that appear in the b-quark jet definitions [6]. The NLO subtraction term for the latter with the same cone radius R is obtained by substituting the vector n J2 forξ J1 in S (1) . After this subtraction, we have to which we can further impose the condition 4(ξ J2 · n J2 ) 2 /n 2 J2 = R 2 for defining a quark (or gluon) jet [7]. Because the thin jet J 1 contributes only the overall normalization inJ E H , the choice of n J1 is arbitrary. We then utilize this freedom, and choose n J1 such that S so the soft function in this special scheme is given by S(ω, R, µ 2 f ) ≈ δ(ω). Equation (3)  600 GeV [12]. In the analysis we include the effects of initial-state and final-state radiations, hadronization, and beam remnants, but turn off the effects from multiple parton interactions. Similar agreement is also observed, cf. Fig. 2, for these boosted electroweak bosons at 1 TeV, though the deviation at r = 0.1 is larger. From these figures, we can get a rough estimate of the effective radii needed to capture all of the radiation for these boosted electroweak bosons. We find that the effective radius needed is R ≈ 0.7 and R ≈ 0.3 for the Higgs boson at 500 GeV and 1 TeV, respectively, and 0.4 and 0.25 for either W or Z bosons. Moreover, the W and Z JEPs are thinner than the Higgs JEP due to their smaller masses. This is further demonstrated via the energy dependence of the JEP for a fixed r value (r = 0.2) in Fig. 3: the separation of the W , Z, and Higgs jets becomes more difficult as the jet energy increases, because the ratio of the mass to the jet energy becomes smaller. Note that the simple expression in Eq. (11) is derived under the aforementioned approximations, at the LO accuracy for the hard kernels, and with the neglect of soft links among the heavy-boson jet and other subprocesses, such as beams and other final-state particles. The associated theoretical uncertainties can be reduced by taking into account relevant corrections, and will be addressed in a future work. Furthermore, the questions of how techniques like trimming [13], pruning [14,15], soft-drop [16,17], and other similar techniques affect the JEP, how subleading corrections affect the JEP, and what happens to the profile if we loosen some of the aforementioned approximations will be investigated in a future work.
In conclusion, we have applied the pQCD factorization to formulate the JEP of a boosted colorless heavy-particle (such as W , Z, Higgs, W ′ and Z ′ boson) jet, which is found to differ dramatically from the ordinary QCD JEPs with similar energy and jet radius. The formalism is greatly simplified by considering inside the boosted jet a thin jet and a fat jet, which absorbs all the soft-gluon effect, when the heavy-particle decays into a quark and antiquark pair. More interestingly, the analytical expression for the JEPs of the electroweak bosons allows for an easy and efficient discrimination of different production processes for these boosted jets. The implementation of this discrimination method can further help suppress QCD background to signals of boosted W and Z jets, after applying conventional kinematic selections.