# Factorization of Jet Mass Distribution in the small limit

###### Abstract

We derive a factorization theorem for the jet mass distribution with a given for the inclusive production, where is a large jet transverse momentum. Considering the small jet radius limit we factorize the scattering cross section into a partonic cross section, the fragmentation function to a jet, and the jet mass distribution function. The decoupled jet mass distributions for quark and gluon jets are well-normalized and scale invariant. And they can be extracted from the ratio of two scattering cross sections such as and . When , the perturbative series expansion for the jet mass distributions works well. As the jet mass becomes small, the large logarithms of appear, and they can be systematically resummed through more refined factorization theorem for the jet mass distribution.

## I Introduction

Jets, collimated bunches of hadrons, contain valuable information to study QCD and high energy interactions. Because jets are well localized in a certain direction, they are rather easily measurable. Also insensitiveness to long distance strong interactions enables us to handle this phenomena perturbatively. Therefore, through comparison between theoretical predictions and experiments, we are able to understand high energy interactions and explore new physics in a keen accuracy.

As the energy of collisions increases, it is useful to employ small radius jets in order to resolve highly energetic particles into multiple jets. This helps us separate signals we are interested in from backgrounds. Also, with small radius jets we can suppress contaminations arising from the underlying/pile-up events.

In a theoretical aspect, the phenomena of a jet with small raidus can be well decoupled from hard collision interactions and systematically described by collinear and (collinear-)soft interactions. Also we can effectively ignore or detach soft gluon emissions with a wide angle from the jet direction. However, the small radius induces large logarithms in the perturbative calculation of . Hence we have to resum the large logarithms to all orders for reliable predictions Dasgupta:2014yra (); Dasgupta:2016bnd (); Kang:2016mcy (); Dai:2016hzf ().

Jet mass is one of the most important jet substructures. When a boosted heavy particle such as a top or Higgs boson decays into a jet, we can identify the heavy particle from the peak of the jet mass if we can separate QCD jets effectively. Therefore a precise description of QCD jet mass distribution is prerequisite for the identification of the heavy particle and moreover understanding physics at TeV scales.

So far the QCD jet mass distributions Banfi:2010pa (); Dasgupta:2012hg (); Chien:2012ur (); Jouttenus:2013hs (); Kolodrubetz:2016dzb () have been widely studied focusing on resumming large logarithms of the small jet mass compared to large or , where is the jet energy and is the jet transverse momentum with respect to an incoming beam axis.
In the small limit a typical scale for the jet mass is comparable to or . So, for example, if is a few TeV, we can describe the jet mass up to a few hundreds of GeV employing the small approximation.^{1}^{1}1For practical use, the size of needs not to be too small.
It is known that the small approximation practically works well even for the case when is 0.6 - 1 Jager:2004jh (); Dasgupta:2012hg ().
However, in the perturbative expansion for QCD jet, the distribution of the nonzero jet mass is
roughly given as . Thus the region is dominant and the resummation of the large logarithms such as Kolodrubetz:2016dzb () is inevitable.
In this case we have to explore QCD dynamics setting the scale hierarchy as .

In this paper, for the inclusive jet production, we introduce the normalized jet mass distribution, which can be universally applied to any isolated QCD jet with small . With the help of the fragmentation functions to a jet (FFJs) Kang:2016mcy (); Dai:2016hzf () we show that the distribution can be separated from the jet scattering cross section with a given or up to next-to-leading order (NLO) in . The jet mass distribution turns out to be scale invariant and can describe the peak region as well as the tail region. We also derive a more refined factorization theorem in focusing on the peak region. This allows us to handle large logarithms arising from the small jet mass compared to .

## Ii Factorization theorem for the jet mass distribution

### ii.1 Collinear Factorization for Jet Substructure

When we consider the inclusive jet scattering cross section such as and the observed jet has a small radius , we can factorize the scattering cross section as Dasgupta:2014yra ()

(1) |

Here is the cross section with a parton in the final state and is the maximal for a given rapidity . Since we consider a jet in the central rapidity region, the rapidity is given to be . is the so-called fragmentation function to a jet (FFJ) Kang:2016mcy (); Dai:2016hzf () for a given mother parton . The FFJ is described as the probability that the outgoing jet from the mother parton acquires a large momentum fraction .

For detailed perturbative results for a jet, throughout this paper we consider anti- algorithm Cacciari:2008gp () for the clustering. At NLO in , for -type algorithms to include Catani:1993hr (); Ellis:1993tq (), anti-, and Cambridge/Aachen (C/A) Dokshitzer:1997in (), the jet merging conditions of two particle emissions are given as

(2) |

Here is the angle between the two particles, and for the hadron collider is assumed to be small and can be approximated by . In our computation as we will see, the typical scale for the observed jet is given as . This is expressed as for annihilation and for hadronic collision.

Based on Eq. (1), we can also investigate substructures of the observed jet. For example, if we try to figure out the fragmentation to a hadron or subjet inside the jet we can employ the following factorization theorem Dai:2016hzf ():

(3) |

where is the hadron or subjet inside , denotes a primary parton flavor for the observed jet, and is a momentum fraction of over . is the so-called jet fragmentation function (JFF), which describes a fragmenting process inside .

In Eq. (3), the FFJs and JFFs are normalized as one and satisfy the momentum conserving sum rule such as

(4) |

Note that the JFFs, , have a limited phase space since whole partons should radiate only inside the jet. To describe the fragmenting process using soft-collinear effective theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001yt (); Bauer:2002nz (), we introduce the so-called unnormalized JFF such as

(5) | |||||

Here , and the collinear quark is given by , where is a collinear Wilson line in SCET Bauer:2000yr (); Bauer:2001yt (). If we consider a gluon jet as an initial state, the fragmentation can be similarly described by a collinear gluon field strength , where is the collinear Wilson line in the adjoint representation. We decomposed the momentum as , where and . Here is an unit vector in the jet direction and is a transverse momentum to . The lightcone vectors and satisfy and .

The definition of the unnormalized JFF in Eq. (5) are almost the same as the usual fragmentation function (FF). The only difference is that the final states for the unnormalized JFF should be inside the jet with a size , while the usual FF has not such a restriction. So, as computed in Refs. Procura:2011aq (); Dai:2016hzf () at NLO, the normalization of differs from one due to a limited phase space and it is given as a function of such as

(6) |

We call it as ‘the integrated jet function (inside a jet)’, which describes parton radiations inside a jet. Therefore the normalized JFF shown in Eqs. (3) and (4) is obtained from dividing the unnormalized JFF by the integrated jet function such as

(7) |

This integrated jet function is also needed for describing the ‘in-jet’ contributions to the FFJs, which have the following structure:

(8) |

where the jet splitting kernel can be expressed as

(9) |

Here is the jet splitting (‘out-jet’) contributions to the FFJs. This factorization for the FFJs in Eq. (8) works at NLO in . It might hold at the higher order if we ignore splitting processes. In deriving Eq. (3) we can use the fact that the fragmentation from to can be factorized as the out-jet and in-jet splitting processes such as

(10) |

Then we multiply/divide the integrated jet function, and finally obtain the factorization theorem in Eq. (3).

In Eq. (3)
if the momentum fractions and are and not too close to 1, we can genuinely describe the FFJs and JFFs by a collinear mode. And we can successfully suppress the contributions from (collinear-)soft degrees of freedom^{2}^{2}2Here the soft degrees of freedom can be separated as a regular soft mode and a collinear-soft mode. The former scales as and the latter as , where and are relevant small parameters to physical situations. The regular soft mode does not contribute to the computation of the jet in the small limit since it cannot recognize the jet boundary and describe radiations only far outside the jet. The collinear-soft mode Bauer:2011uc (); Procura:2014cba (); Becher:2015hka (); Chien:2015cka () describes radiations of boosted soft particles near the jet boundary and can contribute if a jet observable is sensitive to a small momentum. For a detailed decoupling procedure of the collinear-soft mode from the collinear mode we refer to Refs. Bauer:2011uc (); Dai:2017dpc ().
,
which can be decoupled from the collinear mode.
Here the collinear mode scales as ,
where .
Hence it recognizes the jet boundary and gives nonvanishing results for both in-jet and out-jet contributions.
If or in Eq. (3) are close to 1, the collinear-soft contributions do not vanish and the decoupled collinear-soft mode is responsible for radiations with momentum or Dai:2017dpc ().

### ii.2 Factorization to Jet Mass Distribution in the Tail Region:

From now we consider the factorization into a jet mass distribution starting from Eq. (3). In the tail region of the jet mass distribution, the jet mass is comparable with the jet size . In this case the collinear mode with is enough for describing the jet mass distribution, and the collinear-soft contributions are suppressed. In order to incorporate the jet mass distribution with Eq. (3), we introduce fragmenting jet function (FJF) inside a jet Procura:2011aq () putting into the collinear operator for the JFF such as

(11) | |||||

At leading order (LO) in , the FJF is normalized as . The difference compared to a generic FJF Procura:2009vm (); Jain:2011xz (); Ritzmann:2014mka () is that the upper limit of the jet mass is constrained by a jet algorithm, while the generic FJF does not have such a constraint. Then we have the relation

(12) |

where is the maximal jet mass for a given . In case of -type algorithms is given by , where and .

Thus, from Eq. (12), the differential cross section to include information on the jet mass can be written as

(13) |

If we apply momentum sum rule over the final states , we also obtain

(14) | |||||

Here is our desired jet mass distribution for a given and , and written as

(15) |

As seen from Eqs. (12) and (14), is scale invariant except the dependence of in the perturbative series. In Eq. (14), the convolution of and the FFJs is already given to be scale invariant since the renormalization behavior of the FFJs follow DGLAP evolutions resumming large logarithms of small Dasgupta:2014yra (); Dasgupta:2016bnd (); Kang:2016mcy (); Dai:2016hzf (). Also is well-normalized to satisfy . This fact can be clearly seen if we apply the momentum sum rule to Eq. (12). Since is decoupled from the hard scattering process (and the FFJs) as shown in Eq. (14), it can be universally determined for a given jet with and , and can be also applied to annihilation and deep inelastic scattering.

At NLO in , with the clustering condition in Eq. (2) applied, the normalization factor in Eq. (15), i.e, the integrated jet functions , are computed as Cheung:2009sg (); Ellis:2010rwa (); Chay:2015ila ()

(16) | |||||

(17) |

where , , , and is the number of quark flavors.

Computing the FJFs in Eq. (15) and dividing Eqs. (16) and (17), we obtain the normalized jet mass distributions at NLO such as

(18) | |||||

(19) |

where , and is the maximum of the jet mass with anti- algorithm applied. is the so-called “-distribution”, which is defined as

(20) |

where is an arbitrary smooth function at .
In computing Eqs. (18) and (19), the one loop contributions to for are cancelled by at one loop. Then we obtain the scale invariant jet mass distributions, that is also normalized as one.^{3}^{3}3In computing to obtain the jet mass distributions, we applied the zero-bin subtraction Manohar:2006nz () to eliminate the overlap with soft contributions. Here the subtracted contribution does not come from the regular soft mode but the collinear-soft mode scaling as . As shown in Ref. Chien:2015cka (); Chay:2015ila (), the different zero-bin subtraction by the collinear-soft mode gives a different renormalization behavior from the standard jet function that was defined in Eq. (24).

As far as , there is no large logarithm in Eqs. (18) and (19). However, as goes to zero, the large logarithm, , appears. In order to see thee small jet mass behaviors of , we need to use a small value of for the -distribution instead of using . From Eq. (20), using the relation

(21) |

we can rewrite in Eqs. (18) and (19). Here in the right side scales as . Then we can safely take a limit on , and obtain

(22) | |||||

(23) | |||||

So we clearly see the large logarithms of in the small jet mass limit, that needs to be resummed to all order in . For this we need more IR sensitive modes to be decoupled from the collinear mode. In the next section, including these modes we will consider the factorization theorem for the small jet mass distributions.

## Iii Factorization of the Jet Mass Distribution in the Peak Region:

If we assume , the collinear mode with cannot radiate inside a jet since its contribution to the jet mass squared is given to be . Therefore the collinear contributions are involved only in the normalization factor when we consider the distribution . Then the FJF in Eq. (15) should be described by the narrower collinear mode. We will call it ‘ultracollinear mode’, and its scaling behavior is given as satisfying . Since this mode is too narrow to be aware of the jet boundary, the FJF can be identified as a generic FJF without the constraint of the boundary.

Hence if we apply the momentum sum rule to in the ultracollinear limit, it ends up as ‘the standard jet function’ such as Procura:2009vm (); Jain:2011xz ()

(24) |

where is the ultracollinear contribution to the jet mass, and we suppressed the term in the argument of since it does not appear in the ultracollinear limit. is the standard jet function that was first introduced in Ref. Bauer:2001yt (). And, for example, the quark jet function is defined as

(25) |

where the jet function at LO is normalized as .

Because the jet mass is small, it can be also sensitive to collinear-soft radiations. As discussed before, the decoupled collinear-soft mode from the collinear mode does not contribute when . However, if , the momentum squared of the ultracollinear and the collinear-soft modes can be comparable to the small jet mass such as . In general the scaling behavior of the collinear-soft momentum can be written as , where and are small parameters. Here is given by in order that the collinear-soft mode contribute to the jet mass inside the jet. Also using the fact , we estimate . Since we are now interested in the region , should not be much less than , otherwise the small parameter could be . As a result the small parameter is given as , and finally the collinear-soft momentum scales as

(26) |

Note that this collinear-soft mode can read the jet boundary properly like the collinear mode .

From the collinear mode, we can decouple the collinear-soft interactions following the similar procedure performed in Ref. Dai:2017dpc (). Then the decoupled collinear-soft interactions can be expressed in terms of the collinear-soft Wilson lines and , which have the usual form of the soft Wilson lines Bauer:2001yt (); Chay:2004zn () such as

(27) |

Finally, incorporating the collinear-soft interactions with Eq. (15) and applying Eq. (24), we obtain the factorized result of the jet mass distribution for the region such that

(28) | |||||

where the collinear function is equal to , and are the standard jet function introduced in Eq. (24). are the collinear-soft functions, and for it is defined as

(29) |

where is the derivative operator extracting the momentum . The collinear-soft function initiated by a gluon jet can be also defined similarly in the adjoint representation with a normalization factor . At leading order in , is normalized as . From the argument of the delta function in Eq. (29), we see that, at the higher order in , returns nonzero value only from the in-jet contributions, while the out-jet contributions are proportional to . in the last line of Eq. (28) has the dimension , and it can be related as with .

For one collinear-soft gluon emission inside a jet, the phase space constraint for the collinear-soft momentum is given by

(30) |

Employing this we computed the collinear-soft function at NLO and the results are shown as

(31) |

where are for and for . The results are infrared (IR) finite, and the part with has been expressed through the -distribution defined in Eq. (20). The value of in the distribution of Eq. (31) can be arbitrary since the dependence of in the distribution can cancel when combined with the part proportional to . But, the scaling behavior here can be considered as . From Eq. (31) we read the collinear-soft scale to minimize the large logarithms as . This coincides with the fact .

In the factorization theorem in Eq. (28), the standard jet functions for ultracollinear interactions are expressed at one loop such as

Now we have obtained all ingredients for NLO computation of in the framework of factorization. Putting NLO results for (Eqs. (16) and (17)), (Eq. (31)), and (Eqs. (III) and (III)) into Eq. (28), we easily reproduce the results in Eqs. (22) and (23), that are the asymptotic distributions when the small jet mass limit is taken into account in the calculation with the collinear mode.^{4}^{4}4Since the ultracollinear and collinear-soft modes can be considered as subsets of the collinear mode, we can regard the collinear mode as a ‘full mode’ in some sense. So the jet mass distribution with the collinear mode for can cover the full range of the jet mass although we need the resummation of the large logarithms of in the small jet mass region.

## Iv Resummation for the jet mass distribution in the small jet mass region

As we have seen from Eqs. (22) and (23), the results for the small jet mass at the fixed order in is not enough for the precise estimation since the perturbative expansion is not reliable due to the large logarithms. So using the factorization theorem established in Eq. (28) we have to resum the large logarithms to all order in .

The factorized parts , , and in Eq. (28) are governed by the collinear, ultracollinear, and collinear-soft modes respectively. So the relevant scale to each factorized part is given as , , and respectively. Also these scales minimize large logarithms in their own perturbative results.

Resumming procedure from the factorization theorem is given as follows: We factorize at a certain scale, i.e., the factorization scale . Then each factorized part is computed at its own scale to minimize large logarithms. Finally we evolve each factorized part from its own scale to solving renormalization group (RG) equations. Since this procedure does not allow large logarithms in each factorized part, it automatically resums the large logarithms of over all through RG evolutions of , , and .

The anomalous dimensions for the factorized parts satisfy the following RG equations:

(34) |

where . From NLO results, the leading anomalous dimensions are written as

(35) | |||||

(36) | |||||

(37) |

where , , and are for and for