$\bm{N}$-jettiness in electroweak high-energy processes

The understanding of high-energy scattering has reached a state of the art with the advent of the effective theories such as soft-collinear effective theory (SCET) Bauer:2000ew ; Bauer:2000yr ; Bauer:2001ct ; Bauer:2001yt . The basic picture to this understanding lies in the following procedure. The partons from the incoming protons, as in Large Hadron Collider, participate in the hard scattering and produce a plethora of final-state particles including hadrons, electroweak gauge bosons, Higgs particles and possibly heavy particles beyond the Standard Model. Quantum chromodynamics (QCD) plays a major role in comprehending collider physics phenomenology because the strong interaction acts in every stage of the scattering process with disparate energy scales.

The essence of disentangling the strong interaction is to construct factorization theorems which decompose the high-energy processes into hard, collinear and soft parts. SCET is the appropriate effective theory of QCD for high-energy processes, in which energetic, collinear particles form jets immersed in the sea of soft particles. In SCET we select the relevant collinear and soft modes and integrate out all the other degrees of freedom. Factorization of high-energy processes can be accomplished in SCET by decoupling the soft interaction from the collinear particles. Subsequently the collinear sectors in different directions do not interact with each other. Depending on the observables of interest, the phase space is divided by the definite properties of the specific modes and the factorization theorem is constructed according to how the modes or the phase spaces are organized. Each mode has its own characteristic scale, and typically there is a hierarchy of such scales. The scattering cross sections or event shape observables are expressed in terms of the logarithms of the ratios of the different energy scales, which necessitates the resummation of the large logarithms. Because each factorized part is governed by a single scale in each phase space, the large logarithms in each sector can be resummed to all orders using the renormalization group (RG) equations.

So far, we have described the factorization of high-energy processes in QCD. Here we change gears to employ SCET in extremely high-energy electroweak processes, in which all the masses of the particles including the weak gauge bosons can be regarded as small. By way of illustration, we consider the process $e^{-}e^{+}\rightarrow\mu^{-}\ \mathrm{jet},\mu^{+}\ \mathrm{jet}+X$, where $X$ denotes arbitrary final states, and the $\mu$ jet denotes the jet which includes the muon in the final state.¹¹1From now on, we will write $e^{-}e^{+}\rightarrow\mu^{-}\mu^{+}+X$ for the jets including muons. If we refer to the muons instead of the muon jets, we will explicitly state it. In order to simplify the situation, we consider only the weak $SU(2)_{W}$ gauge interaction by turning off all the other gauge interactions. Therefore we imagine a world with the $SU(2)_{W}$ weak gauge interaction out of the full $SU(3)_{C}\times SU(2)_{W}\times U(1)_{B}$ gauge interactions of the Standard Model. The extension to the Standard Model is complicated, but straightforward. In the high-energy limit, the incoming electron can be regarded as a collection of the “partons” which consist of leptons, weak gauge bosons. These partons undergo a hard collision and produce final-state energetic particles, along with soft particles.

This scenario may sound rather dull because everything mimics the processes in QCD, and one may wonder what, if any, can be learned from this imaginary world. The most interesting issue in this context is that the weak gauge interaction is not confining. It means that the incoming particles or the outgoing particles do not have to be gauge singlets. This is in contrast to QCD, where all the strongly-interacting particles are produced as color singlets, that is, hadrons. Due to this constraint, various matrix elements of the operators in QCD in, say, the parton distribution functions (PDF) or the jet functions are evaluated only for the color singlet configurations. However, the gauge singlets, as well as the gauge nonsinglets participate in weak high-energy processes. It makes the procedure of the factorization more sophisticated, and requires more care in analyzing the interwoven structure. Put it in a different way, it corresponds to imagining QCD without confinement and ask how the factorization works if there are free quarks and gluons. The underlying hard scattering processes can be traced directly and the measurement of the jets and the properties can be reconstructed explicitly by the constituents without worrying about hadronization.

We sketch electroweak high-energy processes by analogy with QCD. The incoming on-shell “partons” (electrons in our case) possess certain fractions of the energy from the initial particles and they radiate away gauge bosons to be far off-shell. This process is described by the electroweak beam functions. Then the energetic partons undergo a hard scattering and the final-state particles can be observed in terms of individual particles, described by the fragmentation functions, or jets by the jet functions or the fragmenting jet functions (FJF). And the soft interaction is interspersed between the collinear parts. In electroweak high-energy processes, gauge singlets and nonsinglets are involved in all these factorized components, which makes the analysis more intriguing. This study will be relevant in high-energy electron-positron colliders such as CEPC dEnterria:2016sca , ILC Djouadi:2007ik , FCC-ee Abada:2019zxq , and CLIC Charles:2018vfv . Our analysis can be extended to include the production of the Higgs boson, weak gauge bosons and top quarks.

There appear many different energy scales, the energy of the hard scattering $Q$, the invariant masses of the initial- and final-states as a typical collinear scale, and the soft scale, and possibly more if we are interested in more differential processes describing event shapes. In addition, the mass of the weak gauge bosons $M$ also enters into the picture as a physical mass. In radiative corrections, there are logarithms of the ratios of these scales and we need to resum the large logarithms to all orders. In SCET, the factorization is achieved by dissecting the phase space and devising the modes in that phase space such that the radiative corrections depend on a single scale in each phase space. Then the resummation is obtained by solving the RG equation.

Besides the conventional RG behavior, there is additional rapidity divergence because the phase space is divided into different regions. It is obviously absent in the full theory because there is no separation of the phase space. Therefore it is a good consistency check for an effective theory to see whether the rapidity divergences are cancelled when all the factorized contributions are added. It is one of the goals in this paper to check this point even in the presence of the nonsinglets participating in the scattering.

Though the sum of the rapidity divergences cancels, the rapidity divergence remains in each sector and it affects the RG behavior of the factorized parts. The rapidity divergence has been regulated using diverse methods, such as the use of the Wilson lines off the lightcone collins_2011 , the $\delta$-regulator Idilbi:2007ff ; Idilbi:2007yi , the analytic regulator Becher:2011dz , the rapidity regulator Chiu:2011qc ; Chiu:2012ir , the exponential regulator Li:2016axz , and the pure rapidity regulator Ebert:2018gsn , etc.. Recently one of us has proposed a consistent scheme of applying rapidity regulators to the soft and collinear sectors Chay:2020jzn . It correctly produces the directional dependence in the soft function, which is essential in our case because there are four different collinear directions involved. The independence of the rapidity scale in the cross section critically depends on the interplay between the collinear and the soft functions.

The dependence on the rapidity scale in each sector becomes highly nontrivial and more interesting when gauge nonsinglets as well as singlets participate in high-energy scattering. First of all, let us describe the rapidity divergence for singlets, which is well understood in QCD. The rapidity divergence appears distinctively in $\mathrm{SCET_{I}}$ and $\mathrm{SCET}_{\mathrm{II}}$. In $\mathrm{SCET_{I}}$, since collinear and soft particles have different offshellness, there is no overlap in rapidity between these modes, hence no rapidity divergence arises. In other words, the rapidity divergence cancels in each sector. On the other hand, in $\mathrm{SCET}_{\mathrm{II}}$, where collinear and soft particles have the same offshellness, they overlap near the rapidity boundary. The soft particles with small rapidity cannot perceive the large rapidity region which belongs to collinear sector, causing the rapidity divergence in the soft sector. In the collinear sector, according to our regularization scheme of the rapidity divergence, which will be described in detail, the rapidity divergence arises from the zero-bin subtraction Chiu:2011qc ; Chiu:2012ir . The zero-bin subtraction replaces the spurious rapidity divergence in the naive contribution. It is analogous to the pullup mechanism Manohar:2000kr ; Hoang:2001rr in the dimensional regularization, in which the IR divergence is replaced by the UV divergence. The rapidity divergence may survive in each sector, but their sum cancels. However, the persistent existence of the rapidity divergence in each sector offers additional evolution to complete the resummation.

In weak interaction, the structure of the rapidity divergence is more intricate. For gauge singlets, there is no rapidity divergence as in QCD. For gauge nonsinglets, regardless of $\mathrm{SCET_{I}}$ or $\mathrm{SCET_{II}}$, the rapidity divergence is not cancelled in each sector when the weak charges of the initial or final states are specified. Nor are the Sudakov logarithms. The non-cancellation of the electroweak logarithms is known as the Block-Nordsieck violation in electroweak processes Ciafaloni:2000df ; Ciafaloni:2001vt ; Ciafaloni:2006qu ; Manohar:2014vxa . In contrast, the Sudakov logarithms in QCD cancel from virtual and real contributions in inclusive processes. It yields, for example, the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution of the PDF. The non-cancellation in weak interaction affects the UV behavior as well as the rapidity behavior. Our result is, in some sense, a manifestation of the non-cancellation in considering the event shape through the jettiness.

In the framework of SCET, it was pointed out in ref. Manohar:2018kfx that the nonsinglet electroweak PDFs possess this property. It is also true for the beam function in the initial state, and for the jet functions, or the FJFs in the final state, and for the soft function. The purposes of this paper are to analyze the structure of the factorization in the weak processes, and to resum large logarithms by probing the structure of the divergences including the rapidity divergence from the nonsinglet contributions.

As a concrete example, we consider an event shape, especially the $N$-jettiness Stewart:2010tn ; Jouttenus:2011wh (in fact, 2-jettiness) in $e^{-}e^{+}\rightarrow\mu^{-}\mu^{+}+X$. By considering the jettiness, we can probe the hierarchy of scales in effective theories, and the characteristics of a measurement-dependent factorization can be discussed. In addition, we are interested in the case in which there are four distinct lightlike directions. For this reason, $N$-jettiness is better suited than the beam thrust in the Drell-Yan process Stewart:2010pd or the jet thrust in $e^{-}e^{+}$ collisions. In terms of QCD, it corresponds to the $N$-jettiness from the partonic subprocess $q\overline{q}\rightarrow q^{\prime}\overline{q}^{\prime}+X$.

The rest of the paper is organized as follows. In section 2, we explain the $N$-jettiness, and establish the power counting of the relevant momenta and the jettiness to choose the appropriate effective theories. In section 3, we first construct the factorization of the $N$-jettiness in $\mathrm{SCET_{I}}$, in which the ingredients of the factorized parts are extracted and defined. The beam functions, the semi-inclusive jet functions, and the soft functions are constructed in the context of $\mathrm{SCET_{I}}$ with the weak singlet and nonsinglet contributions. Then we scale down to $\mathrm{SCET_{II}}$  and establish the factorization by probing the relations between the beam function in $\mathrm{SCET_{I}}$, and the PDF in $\mathrm{SCET_{II}}$, or those between the fragmenting jet functions and the fragmentation functions. In section 4, we examine the source of the rapidity divergence, set up the rapidity regulators in the collinear and the soft sectors and discuss their characteristics.

In section 5, we present the radiative corrections of the collinear parts at next-to-leading order (NLO), which consist of the beam functions, the PDFs, the jet functions, the fragmentation functions and the FJFs. In section 6, the hard functions are collected for the process $\ell_{e}\overline{\ell}_{e}\rightarrow\ell_{\mu}\overline{\ell}_{\mu}$ with the left-handed electron and muon doublets, and the hard anomalous dimension matrix is presented. The soft functions are analyzed in section 7, which are expressed in terms of the matrices in weak-charge space. In section 8, we combine all the factorized parts to show the RG evolution of the $N$-jettiness, and we conclude in section 9. Long technical calculations are relegated to appendices. In appendix A, we list the Laplace transforms of the distributions. In appendices B and C, we present how to obtain the collinear functions and the matching coefficients in the limit of small $M$. In appendix D, we illustrate the color matrices for the soft functions at tree level, and we show that there is no mixing at one loop for the soft functions with four nonsinglets.

We consider the $N$-jettiness²²2We keep using the terminology $N$-jettiness, bearing in mind that we actually consider the 2-jettiness $\mathcal{T}_{2}$ in our case., which is defined as Stewart:2010tn ; Jouttenus:2011wh

where $i$ runs over 1, 2 for the beams, and $3,\cdots,N+2$ for the final-state jets. Here $q_{i}$ are the reference momenta of the beams and the jets with the normalization factors $\omega_{i}=\overline{n}_{i}\cdot q_{i}$, and $p_{k}$ are the momenta of all the measured particles in the final state.

where $z_{1,2}$ are the momentum fractions of the beams. Here $n_{1}$ and $n_{2}$ are lightcone vectors for the beams, which are aligned to the $z$ direction, $n_{1}^{\mu}=(1,0,0,1),\ n_{2}^{\mu}=(1,0,0,-1)$, and $n_{i}$ are the lightcone vectors specifying the jet directions. Typically we write the lightlike vectors as $n_{i}=(1,\mathbf{n}_{i})$ , $\overline{n}_{i}=(1,-\mathbf{n}_{i})$, where the unit vector $\mathbf{n}_{i}$ denotes the direction of the spatial momentum $\mathbf{p}_{i}$. The $N$-jettiness in eq. (1) can also be expressed in terms of some arbitrary hard scales instead of the normalization factor $\omega_{i}$.

In references Jouttenus:2011wh ; Bertolini:2017efs , the authors have considered the differential distribution with respect to the individual jettiness, which is defined as

Here we consider the total $N$-jettiness $\mathcal{T}_{N}$, which is given by $\mathcal{T}_{N}=\sum_{i}\tilde{\mathcal{T}}_{i}$.

For $\mathcal{T}_{2}\ll Q$, SCET can be applied to the process $e^{-}e^{+}\rightarrow\mu^{-}\mu^{+}+X$, but we should determine which effective theories are to be employed, depending on the hierarchy of the scales in the collinear and soft momenta and the magnitude of the jettiness. The $n$-collinear momentum scales as $p_{c}^{\mu}=(\overline{n}\cdot p_{c},p_{c\perp},n\cdot p_{c})=(p_{c}^{-},p_{c\perp},p_{c}^{+})\sim Q(1,\lambda,\lambda^{2})$, where $\lambda\sim p_{c\perp}/\overline{n}\cdot p_{c}$ is the small parameter in SCET. We can consider either $\mathrm{SCET_{I}}$, or continue down to $\mathrm{SCET_{II}}$, depending on the power counting of the soft momentum.

In $\mathrm{SCET}_{\mathrm{I}}$, the ultrasoft (usoft) momentum scales as $p_{us}^{\mu}=(p_{us}^{-},p_{us\perp},p_{us}^{+})\sim Q(\lambda^{2},\lambda^{2},\lambda^{2})$, while the soft momentum in $\mathrm{SCET}_{\mathrm{II}}$ scales as $p_{s}^{\mu}=(p_{s}^{-},p_{s\perp},p_{s}^{+})\sim Q(\lambda,\lambda,\lambda)$. The gauge boson mass $M$ is another mass scale which enters into the system. When the usoft or the soft gauge bosons are on their mass shell, $p_{us}^{2}\sim p_{s}^{2}\sim M^{2}$, it implies that the gauge boson mass $M$ is power counted as $Q\lambda^{2}$ in $\mathrm{SCET_{I}}$, and $Q\lambda$ in $\mathrm{SCET_{II}}$.

The scale of the $N$-jettiness $\mathcal{T}_{N}$ from eq. (1), is extracted from the lightcone component $p^{+}$. Note, however, that the magnitudes of $p^{+}$ for the collinear and the usoft momenta in $\mathrm{SCET_{I}}$ are comparable to each other, hence the contribution to the $N$-jettiness comes both from the collinear and the usoft sectors. On the other hand, the magnitude of $p^{+}$ for the collinear momentum is much smaller than that for the soft momentum in $\mathrm{SCET_{II}}$. In this case, the contribution to the $N$-jettiness comes only from the soft sector. Therefore we expect that the structure of the factorization takes different forms in $\mathrm{SCET_{I}}$ and in $\mathrm{SCET_{II}}$.

In $\mathrm{SCET_{I}}$, the hierarchy of the scales is given as $\mathcal{T}^{2}\sim M^{2}\ll p_{c}^{2}\sim Q\mathcal{T}\ll Q^{2}$. The relevant modes scale as

The $N$-jettiness probes the scale of order $p_{c}^{+}\sim p_{us}^{+}\sim Q\lambda^{2}\sim\mathcal{T}$, hence both the collinear and the usoft parts contribute to the $N$-jettiness.

If we consider the kinematical situation in which $M^{2}\ll p_{c}^{2}\sim Q\mathcal{T}$, the framework of $\mathrm{SCET_{I}}$ suffices to describe the $N$-jettiness. However, we would like to include another kinematical case, in which the jet becomes narrower, and we reach the region $M^{2}\sim p_{c}^{2}\sim\mathcal{T}^{2}$. Then the plus component $p_{c}^{+}\sim\mathcal{T}^{2}/Q$ does not contribute to the jettiness. However, there remains large logarithms associated with $Q/M$, which should be resummed. This can be achieved by going from $\mathrm{SCET_{I}}$ to $\mathrm{SCET_{II}}$ through the matching. The beam functions and the jet functions in $\mathrm{SCET_{I}}$ have virtuality $p_{c}^{2}\sim QM$, and we need a second stage of matching to separate these scales. And the ultrasoft function is scaled down to the soft function, which develops the rapidity divergence. These necessitate the use of $\mathrm{SCET_{II}}$.

In $\mathrm{SCET_{II}}$, the hierarchy of scales is given by $\mathcal{T}^{2}\sim M^{2}\sim p_{c}^{2}\ll Q\mathcal{T}\ll Q^{2}$. The hard-collinear, collinear and soft modes scale as

In fact, the hard-collinear modes are not the dynamical degrees of freedom in $\mathrm{SCET_{II}}$, but these are the modes from $\mathrm{SCET_{I}}$ to be integrated out to obtain $\mathrm{SCET_{II}}$. The $N$-jettiness probes the scale of order $p_{s}^{+}\sim Q\lambda\sim M$, and the collinear contribution with $p_{c}^{+}\sim Q\lambda^{2}\sim M^{2}/Q$ does not contribute to the jettiness. This is also recognized in ref. Lustermans:2019plv in a different context of measuring the transverse momentum $q_{T}$ and the 0-jettiness $\mathcal{T}_{0}$.

We first describe $\mathrm{SCET_{I}}$ in order to set up the elements of the factorization. In order to obtain $\mathrm{SCET_{II}}$, the hard-collinear modes, which are previously collinear modes in $\mathrm{SCET_{I}}$, are integrated out to reach the collinear modes in $\mathrm{SCET_{II}}$. Note that the scaling of the usoft momentum in $\mathrm{SCET_{I}}$, and the soft momentum in $\mathrm{SCET_{II}}$ remain the same, but the small parameter $\lambda$, responsible for the power counting, is rescaled from $\sqrt{\mathcal{T}/Q}$ to $\mathcal{T}/Q$. We present the results in both cases.