Resummation prediction on the jet mass spectrum in one-jet inclusive production at the LHC

# Resummation prediction on the jet mass spectrum in one-jet inclusive production at the LHC

Ze Long Liu, School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing 100871,ChinaCenter for High Energy Physics, Peking University,
Beijing 100871,ChinaPRISMA Cluster of Excellence Mainz Institute for Theoretical Physics,Johannes Gutenberg University,
D-55099 Mainz, Germany
Chong Sheng Li, School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing 100871,ChinaCenter for High Energy Physics, Peking University,
Beijing 100871,ChinaPRISMA Cluster of Excellence Mainz Institute for Theoretical Physics,Johannes Gutenberg University,
D-55099 Mainz, Germany
Jian Wang School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing 100871,ChinaCenter for High Energy Physics, Peking University,
Beijing 100871,ChinaPRISMA Cluster of Excellence Mainz Institute for Theoretical Physics,Johannes Gutenberg University,
D-55099 Mainz, Germany
and Yan Wang School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing 100871,ChinaCenter for High Energy Physics, Peking University,
Beijing 100871,ChinaPRISMA Cluster of Excellence Mainz Institute for Theoretical Physics,Johannes Gutenberg University,
D-55099 Mainz, Germany
###### Abstract

We study the factorization and resummation prediction on the jet mass spectrum in one-jet inclusive production at the LHC based on soft-collinear effective theory. The soft function with anti- algorithm is calculated at next-to-leading order and its validity is demonstrated by checking the agreement between the expanded leading singular terms with the exact fixed-order result. The large logarithms and the global logarithms in the process are resummed to all order at next-to-leading logarithmic and next-to-next-to-leading logarithmic level, respectively. The cross section is enhanced by about 23% from the next-to-leading logarithmic level to next-to-next-to-leading logarithmic level. Comparing our resummation predictions with those from Monte Carlo tool PYTHIA and ATLAS data at the 7 TeV LHC, we find that the peak positions of the jet mass spectra agree with those from PYTHIA at parton level, and the predictions of the jet mass spectra with non-perturbative effects are in coincidence with the ATLAS data. We also show the predictions at the future 13 TeV LHC.

###### Keywords:
\arxivnumber

1412.1337 \preprintMITP/14-094

## 1 Introduction

The substructure of jets produced at the Large Hadron Collider (LHC) has become one of the hot topics for both theorists and experimentalists. The particles such as massive electroweak bosons, top quark and other possible new resonances produced with transverse momenta much greater than their masses, i.e., , can decay to hadronic products, which are almost collinear and may be recombined into a single jet by jet algorithms. Therefore it is necessary to find a way to distinguish the interesting signal jets from the purely QCD backgrounds.

During the past few years, many studies on jet substructures have been performed [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], in which new techniques and observables have been designed to analyze the events. The event generators such as SHERPA [14, 15], PYTHIA [16, 17] and HERWIG++ [18, 19], can provide fully differential events, by which any observable can be predicted and compared with data. However, the various event generators employ different models for parton shower and non-perturbative effects, such as the hadronization and multiparton interactions. As a consequence, they might provide very different predictions. For instance, the jet mass spectra from the PYTHIA and the HERWIG++ do not agree with each other, as shown in ref. [20]. Moreover, there is a type of color correlation between the initial and final colored particles that is not taken into account in these event generators.

In order to obtain more precise predictions and test the validity of the Monte Carlo tools, it is important to develop a theoretical framework to study the jet substructure. Recently, various jet substructure observables have been investigated analytically based on soft-collinear effective theory (SCET) [21, 22, 23, 24, 25, 26, 27, 28, 29] and the traditional perturbative QCD (pQCD) resummation formalism [30, 31, 32, 33, 34]. For example, the factorization and resummation prediction of the jet angularity in the multijet production at colliders have been studied in refs. [23, 24], and the invariant mass and energy profile of jets at hadron colliders have been explored in refs. [31, 32].

The theoretical developments of prediction on jet mass spectrum at hadron colliders can be found in [32, 33, 28, 35]. In ref. [32], the jet mass was investigated with the pQCD resummation formalism by focusing on the processes independent jet function, where it was found that the nonperturbative effects are important at small jet mass. The author of ref. [33] studied the distributions of in and processes at NLL, using the formula in refs. [36, 37], and including resummation effects of non-global logrithms (NGLs) in large- approximation. The jet mass spectrum with the process was discussed [35] in the -jettiness global event shape [38]. The factorization formula and resummation prediction of the jet mass spectrum for direct photon production in the framework of SCET was provided in ref. [28], where the soft function was factorized into two pieces with different scales. Thought the non-global logarithms were not resummed there, their contribution were estimated and it was found that the NGLs only affect the jet mass spectrum in the peak regions significantly.

Studies of the jet mass can not only help us understand QCD, but also be useful to search for new physics, especially in the complex QCD environment of the LHC. In particular, if we want to identify the mass peak of a highly boosted particle, the jet mass spectrum of QCD background must be calculated precisely. Actually, the jet invariant mass were explored in both ATALS and CMS collaborations at the 7 TeV LHC [20, 39]. From these results, we can see that the jet mass peaks at about 50 GeV, which can be much smaller than the transverse momenta of jet . Therefore there exist large logarithmic terms with in the perturbative calculations near the peak region, which need to be resummed to all order in order to give reliable predictions.

In this paper, we study one of the simplest jet substructures, i.e. the invariant mass of a jet, and investigate the factorization and resummation prediction on the jet mass spectrum in SCET for one-jet inclusive production at the LHC. Compared with direct photon process [28], the factorization formula for dijet process is more complicated due to the nontrivial color structure and associating soft radiation. The illustrative picture of this process is shown in figure 1. Since the soft radiation can either be inside or outside the cone of the measured jet, there are two kinematic variables which can lead to large logarithms at threshold limit: one is the invariant mass of the measured jet, and another is the invariant mass of the partonic system that recoils against the observed jet. In the threshold region and , both of the large logarithms and need to be resummed to all order. In the threshold limits, the cross section can be factorized as

 σ=fPa⊗fPb⊗H⊗S⊗Jobs.⊗Jrec., (1)

where , , , are the hard function, soft function, jet function and parton distribution function (PDF), respectively. Both of the hard and soft function are matrices in color space. The hard function includes the short distance contributions arising from virtual corrections. The jet function presents the collinear radiation in the jet. The indices “obs.” and “rec.” denote the observed jet and the recoiled one, respectively. The effects from soft gluon emission are incorporated in the soft function and its phase space is constrained by the jet algorithms. It is noteworthy that the large angle soft gluon arising from the initial state radiation (ISR) and recoiled final state radiation are taken into account in this formalism. In contrast to the cone algorithm adopted in ref. [28], we choose anti- algorithm [40] to calculate the jet and soft functions, which is boost-invariant and stable against the change of jet boundary [41]. Thus, our prediction can be valid for the jet with both small and large rapidity, and is more useful for phenomenological purposes.

This paper is organized as follows. In section 2, we analyze the kinematics of the one-jet inclusive production at hadron colliders and give the definition of the threshold region. In section3, we derive the factorization formula. In section 4 and section 5, we show the results of hard function and jet function at NLO, respectively. We calculate the soft function at NLO and present its refactorization in section 6. In section 7, we give the final renormalization group (RG) improved cross section analytically. In section 8, we discuss the numerical results of the jet mass distribution for one-jet inclusive production at the LHC, including the leading singular distribution at threshold limit, scale uncertainties, dependence, distinction between quark jets and gluon jets, and comparison between the RG improved predictions and ATLAS data. We conclude in section 9.

## 2 Analysis of kinematics and factorization

In this section, we introduce the relevant kinematical variables and the factorization formula needed in our analysis. We consider the process

 N1(Pa)+N2(Pa)→J(pJ1)+X, (2)

where denotes the leading final jet, and is its invariant mass. The partonic channels include , , and their various crossing ones. The Feynman diagrams at leading order (LO) are shown in appendix A.

It is convenient to introduce two lightlike vectors and along the beam directions, and another lightlike vector along the measured jet direction. In the center-of-mass (CM) frame of the initial partons, for the one-jet inclusive production, the momentum of recoiling parton to the observed jet is along the direction . In the CM frame of the hadronic collision, the momenta of the incoming hadrons are given by

 Pμa=ECMnμa2,Pμb=ECMnμb2. (3)

Here is the CM energy of the collider and we have neglected the mass of the hadrons. The momenta of the incoming partons, with a light-cone momentum fraction of the hadronic momenta, are

 pa=xaECMnμa2,pb=xbECMnμb2. (4)

The hadronic kinematic invariants are defined as

 s=(Pa+Pb)2,t1=(Pa−pJ1)2−m2J1,u1=(Pb−pJ1)2−m2J1, M2X≡P2X=(Pa+Pb−pJ1)2=s+t1+u1+m2J1, (5)

and the partonic ones are defined as

 ^s=(pa+pb)2=xaxbs,^t1=(pa−pJ1)2−m2J1=xat1,^u1=(pb−pJ1)2−m2J1=xbu1, s4≡m2X=(pa+pb−pJ1)2=^s+^t1+^u1+m2J1, (6)

where . In the threshold limits, we have and . The kinematic region we are interested in is

 ^s,^t1,^u1≫m2J,s4≫Λ2QCD (7)

Any four vector can be decomposed along the light-like reference vector

 pμ=(ni⋅p)¯nμi2+(¯ni⋅p)nμi2+pμ⊥=p+¯nμi2+p−nμi2+pμ⊥. (8)

Hence the momentum can be denoted by . The momentum modes relevant to our discussions are the collinear mode , anti-collinear mode and soft mode , where is treated as a small expansion parameter. In the partonic threshold limits and , the radiation is constrained to be either soft or collinear with the final-state partons.

In order to identify energetic cluster of radiation, the sequential recombination jet algorithms are used. The longitudinal boost invariant distance measures and are defined by

 dij =min(pαT,i,pαT,j)ΔRij/R,ΔRij=√(yi−yj)2+(ϕi−ϕj)2, diB =pαT,i,

where is the jet radius parameter, and are rapidity and azimuthal angle of the jet , respectively. , 0 and 1 represent the inclusive anti- [40], Cambridge-Aachen [42, 43] and  [44, 45] jet algorithms, respectively. The effects of jet algorithms on the resummation have been studied in refs. [46, 47, 48, 41, 49], among which ref. [41] has shown that jet boundary can be changed significantly by boundary clustering for Cambridge-Aachen and algorithms, while the change of the phase space is power suppressed for anti- algorithm. In this paper, the anti- algorithm is adopted, and the jet boundary is just a circle of radius in plane around the jet direction.

After clustering jets, the jet invariant mass receives contribution from the radiation inside the jet, whether from collinear and soft gluons. Thus we split the soft radiation to two parts, denoted by . Then, the partonic threshold variables take the form

 m2J =(pJ1+kin)2=m2J1+2kin⋅pJ1, (9) s4 =(pJ2+kout)2=m2J2+2kout⋅pJ2.

In the kinematic region , the momenta of the two jets can be written as and in the partonic CM frame, where in the threshold limit. And and can be rewritten as

 m2J =m2J1+2EJ(nJ⋅kin), (10) s4 =m2J2+2EJ(¯nJ⋅kout).

For later convenience, we write and .

The hadronic threshold is defined as . In this limit ,the final state radiations and beam remnants are highly suppressed, which leads to final states consisting of two narrow jets, as well as the remaining soft radiations. For convenience, we introduce the dimensionless variables

 v=1+^t1^s,w=−^u1^s+^t1,¯¯¯v=1−v. (11)

In terms of , , and ,

 M2X=m2Xx2+E2CM[(1−x1)v+(1−x2)¯¯¯v]+m2J. (12)

In the limit , , and , we have

 M2X=m2X+m2J+p2Tv¯¯¯v[(1−x1)v+(1−x2)¯¯¯v]+…. (13)

This expression is helpful when we derive the RG equation of the soft function by using the RG invariance in section 6.

## 3 Factorization in SCET

To derive a factorization formula for dijet process in SCET, we first have to match the full QCD onto the effective theory [50, 51]. To illustrate the factorization in detail, we consider the process . The initial partons are labeled by 1 and 2 and the final partons are labeled by 3 and 4, and the relevant operator in QCD is given by [52]

 OQCDIΓ=(¯ψa44γμΓψa22)(¯ψa33γμΓ′ψa11)(cI){a}, (14)

where denotes a 4 order color tensor with color indices , and denote the chirality ( or ). In SCET, the -collinear quark field can be written as

 χn(x) = W†n(x)ξn(x),ξn(x)=n/¯n/4ψn(x), (15)

where is the Wilson line, and is the gauge invariant combination of and collinear quark field in SCET. At the leading power in , only the component of soft gluons can interact with the -collinear field , which can be decoupled by a field redefinition [53]:

 χn(x)→Yn(x)χn(x), (16)

with

 Yn(x)=Pexp(igs∫0−∞dsn⋅Aas(x+sn)ta), (17)

Then the effective Lagrangian can be expressed as

 Leff=∑I,ΓCΓIOSCETIΓ, (18)

with

 OSCETIΓ =∑{a}(cI){a}[Oc(x)]b1b2b3b4Γ[Os(x)]{a},{b}, (19) b1b2b3b4Γ =¯χb4¯nJ(x)γμΓχb2¯n(x)¯χb3nJ(x)γμΓ′χb1n(x), (20) {a},{b} =[Y†¯nJ(x)]b4a4[Y¯n(x)]a2b2[Y†nJ(x)]b3a3[Yn(x)]a1b1. (21)

Here is the hard matching coefficient. The scattering amplitude for the can be written as

 |MΓ(x)⟩=⟨X|OcΓ(x)Os(x)|N1N2⟩|CΓ⟩, (22)

where is the vector of Wilson coefficient combination in color basis , as following

 |CΓ⟩=∑ICΓI|cI⟩. (23)

For , the color basis is chosen as

 |c1⟩=tci3,i1tci4,i2,|c2⟩=δi3,i1δi4,i2. (24)

The differential cross section can be written as

 dσdpTdydm2J=12s∑X∑Γ∫d4x⟨MΓ(x)|ˆM(m2J,pT,y,R)|MΓ(0)⟩, (25)

where the operator denotes the measurement in the final state, including the jet algorithm. It acts on the final-state collinear and soft particles with momenta as follows

 ˆM(m2J,pT,y,R)|Xc+s⟩=M(m2J,pT,y,R,pc,ks)|Xc+s⟩, (26)

where

 M(m2J,pT,y,R,{pc},{ks})= δ((pc+ks)2−m2J)δ(|→pcT|−pT)δ(y−12lnp+cp−c) (27) ×Θ(R2−(ys−yc)2−(ϕs−ϕc)2).

Since the soft and collinear sectors are decoupled due to field redefinition, the matrix element in eq. (25) can be factorized into a product of several matrices,

 ∑X⟨M(x)|ˆM(m2J,pT,y,R)|M(0)⟩= (28) ×⟨N1(P1)|¯χα1n(x)χβ1n(0)|N1(P1)⟩ ×⟨N2(P2)|¯χα2¯n(x)χβ2¯n(0)|N2(P2)⟩ ×∑Xc1⟨0|χγ1nJ(x)|Xc1⟩⟨Xc1|¯χσ1nJ(0)|0⟩ ×∑Xc2⟨0|χγ2¯nJ(x)|Xc2⟩⟨Xc2|¯χσ2¯nJ(0)|0⟩ ×∑Xs⟨CΓ|⟨0|Os†(x)|Xs⟩⟨Xs|Os(0)|0⟩|CΓ⟩ ×M(m2J,pT,y,R,{pc},{ks}),

where denotes the average over the colors and spin of the initial-state partons, and , etc, are Dirac indices. The initial state collinear sectors match to the conventional PDFs:

 ⟨Ni(Pi)|¯χα1i(ni⋅x¯nμi2)χβ1i(0)|Ni(Pi)⟩=12¯ni⋅Pi(n/i2)β1α1∫1−1dξfq/Ni(ξ)eiξ(ni⋅x)(¯ni⋅Pi)/2, (29)

and the matrix elements of the collinear fields in the final state match to the quark jet function:

 ∑Xc1⟨0|χγ1ni(x)|Xc1⟩⟨Xc1|¯χσ1ni(0)|0⟩=(n/i2)γ1σ1∫d4p(2π)3θ(p0)(¯nJ⋅p)Jq(p2)e−ixp. (30)

The soft function can be defined as the matrix element associated with the soft Wilson line

 S(x,μ)=⟨0|Os†(x)|Xs⟩⟨Xs|Os(0)|0⟩, (31)

which can be decomposed in the color basis

 SIJ≡⟨cI|S|cJ⟩. (32)

Now the matrix element appearing in eq. (28) can be simplified as

 ⟨CΓ|⟨0|Os†(x)|Xs⟩⟨Xs|Os(0)|0⟩|CΓ⟩=∑IJCΓ∗ISIJCΓJ. (33)

All the above components in the factorization form in eq. (28) satisfy certain RG equations, which we will discuss in the following sections. Combining the different parts together, we get the factorized differential cross section in the threshold limits

where is the hard-scattering kernel

 Cij(^s,^t1,^u1,m2J,R)= ∑I,J∫dm2J1dm2J2dkindkoutHIJ(^s,^t1,^u1)SJI(kin,kout) (35) ×J1(m2J1)J2(m2J2)δ(m2J−m2J1−2EJkin)δ(s4−m2J2−2EJkout),

with

 HIJ=∑ΓCΓICΓ∗J. (36)

And is the hard function, the details of which are shown in section 4.

For other channels, such as or , the formula of factorization is similar to the process , except for the different jet functions and PDFs. The definitions of gluon PDF and jet function are given by

 ⟨Ni(Pi)|(−gμν)Aμi⊥(ni⋅x¯nμi2)Aνi⊥(0)|Ni(Pi)⟩=∫1−1dξξfg/Ni(ξ)eiξ(ni⋅x)(¯ni⋅Pi)/2, (37)

and

 ⟨0|AaJμ⊥(x)AbJν⊥(0)|0⟩=δab(−gμν⊥)g2s∫d4p(2π)3θ(p0)Jg(p2)e−ixp. (38)

## 4 Hard function

The coefficient can be obtained by matching the full theory onto SCET. The one loop results for all partonic process in QCD have been available in ref. [52], which are derived in dimensional regularization and the renormalization scheme. In this section, we show the crossing relations for different channels and the RG evolution briefly. The explicit expressions of hard matching coefficients are shown in appendix B.

### 4.1 Wilsons coefficient at NLO

First, for the 4-quark processes, there are six channels if two different flavor quarks are involved (e.g. )

 qq′→qq′,q¯q′→q¯q′,q¯q→q′¯q′,qq′→q′q,q¯q′→¯q′q,q¯q→q′¯q′. (39)

The Wilson coefficients for the channel are denoted by and the others can be obtained by crossing symmetries, as shown in table 1. For example, the Wilson coefficients for the channel are .

in Wilson coefficients denotes the chirality of the incoming and outgoing partons. In general, there are 16 possible chirality amplitudes. Actually, for the channel , only 4 chirality amplitudes are non-zero. This is because that chiralities of massless particles 1 and 3 (2 and 4) must be the same. We rewrite the Wilson coefficients as with . In addition, since the chirality can be changed by charge conjugation, the only two independent chirality amplitudes for are and .

If the 4 quarks are identical, there are two additional non-vanishing chirality amplitudes and because of the contribution of u-channel. The interference between t-channel and u-channel also makes the results different from case. The results for can be expressed as

 CLLLLI =CRRRRI=CLLI(s,t,u)+BIJCLLJ(s,u,t), CLRLRI =CRLRLI=CLRI(s,t,u), (40) CLRRLI =CRLLRI=BIJCLRJ(s,u,t),

where

 BIJ =⎛⎜⎝−1CA2−CFCA1CA⎞⎟⎠. (41)

The results of other channel associated with can be obtained by crossing symmetry as shown in table 1.

Next, we consider the Wilson coefficients for channel and its crossing. There are six relevant channels

 gg→q¯q,qg→qg,¯qg→¯qg,gg→¯qq,qg→gq,¯qg→g¯q. (42)

The Wilson coefficients for the channel are denoted by and the others can be obtained by crossing symmetries as shown in table 1. According to parity invariance, we have

 Cλ1,λ2,λ3,λ4I=C−λ1,−λ2,−λ3,−λ4I. (43)

In addition, when . Thus, the Wilson coefficients for can be rewritten as , and there are only 4 independent chirality amplitudes for each color structure, the explicit expressions of which are shown in appendix B.

Finally, we consider the process . In ref. [52], the Wilson coefficients are obtained by matching to an overcomplete basis of 9 color structures, though there are only 8 independent color structures. Then, 16 possible helicity amplitudes for each color structures give 144 matching coefficients. Basing on the symmetry, the Wilson coefficients can be expressed concisely as follows

 CΓI =4g2sMΓI(1+αs4πQΓI) I=1⋯6 Γ=1⋯6, (44) CΓI =4g2sαs4πQΓI I=7,8,9 Γ=1⋯6, CΓI =4g2sαs4πQΓI. I=1⋯9 Γ=7⋯16.

The explicit expressions of and are listed in appendix B for the convenience of the reader.

### 4.2 RG evolution of the hard function

The Wilson coefficients satisfy the RG equation [54, 55, 56, 57, 58]

 ddlnμCΓI(μ)=ΓHIJCΓJ(μ), (45)

where can be expressed as

 ΓHIJ(s,t,u,μ) =(γcuspcH2ln−tμ2+γH−β(αs)αs)δIJ+γcuspMIJ(s,t,u), (46)

with

 cH =nqCF+ngCA, (47)

and

 γH =nqγq+ngγg, (48)

where is the QCD beta function, is the cusp anomalous dimension, and and is the number of external quarks and gluons involved in the process, respectively. denotes the color mixing terms, and can be written as

 M=−∑i≠jTi⋅Tj2[L(sij)−L(t)], (49)

where , , , and is defined as

 L(x)=ln|x|μ2−iπθ(x). (50)

The explicit expressions of for each channel can be found in appendix B. can be diagonalized with eigenvalues . For example, for channel, we have

 (51)

where denotes the transform matrix, which can be calculated numerically. The Wilson coefficients in the diagonal basis are denoted by , which satisfy the RG equation

 ddlnμ^CΓI(μ) =[γcuspcH2ln−tμ2+γH+γcuspλK−β(αs)αs]^CΓI(μ). (52)

The hard function in the diagonal basis is denoted by . With eq. (36), the RG equation of the hard function can be obtained,

 ddlnμ^HKK′(μ)=[γcusp(cHln∣∣∣^t1μ</