MSUHEP-18-018

Resummation of High Order Corrections in Z Boson Plus Jet Production at the LHC

Peng Sun,1, ∗ Bin Yan,2, † C.-P. Yuan,2, ‡ and Feng Yuan3, §

1Department of Physics and Institute of Theoretical Physics,Nanjing Normal University, Nanjing, Jiangsu, 210023, China

2Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA3Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

We study the multiple soft gluon radiation effects in Z boson plus jet production at the LHC.By applying the transverse momentum dependent factorization formalism, the large logarithmsintroduced by the small total transverse momentum of the Z boson plus leading jet final statesystem, are resummed to all orders in the expansion of the strong interaction coupling at theaccuracy of Next-to-Leading Logarithm(NLL). We also compare the prediction of our resummationcalculation to the CMS data by employing a reweighting procedure to estimate the effect fromimposing kinematic cuts on the leptons from Z boson decay, and find good agreement for both theimbalance transverse momentum and the azimuthal angle correlation of the final state Z boson andleading jet system, for pp→ Z + jet production at the LHC.

Introduction. The Z boson and jets associated produc-tion at Large Hadron Collider (LHC) plays an importantrole in our knowledge of the Standard Model (SM) andbeyond. The clean and readily identifiable signature andlarge production rate of this process provide an oppor-tunity to precisely measure the electroweak parameters,constrain the parton distribution functions (PDFs) andalso probe the strong coupling constant αs. In particular,it is a prominent background in searches for SM processesand physics beyond the SM at the TeV scale [1]. There-fore, a precise study of both the inclusive and differentialmeasurements of Z boson plus jets production is vital totest the SM and search for new physics (NP).

Currently, both the ATLAS and CMS collaborationshave reported the measurements of Z boson productionassociated with zero, one and two jets [2–5]. Althoughthe experimental measurements on pT and y of jet showvery good agreement with theoretical predictions [6], abetter theoretical calculation for some other observables(e.g., the total transverse momentum of Z boson andleading jet system) in regions of the phase space domi-nated by soft/collinear radiation are still needed to re-duce the theoretical uncertainties. Both the fixed-orderand resummation techniques could be used to improvethe theoretical predictions. Perturbative QCD correc-tions to the Z boson plus multijets production at thenext-to-leading order (NLO) are widely discussed in lit-eratures [7–14]. The NLO effects from electroweak cor-rection to Z boson plus multijets are also discussedin Refs. [15–18]. Beyond the NLO QCD calculation,the leading threshold logarithms have been included inRef. [19]. The accuracy to the Z boson plus one jet pro-duction has reached to the next-to-next-to-leading or-

∗ pengsun@msu.edu† yanbin1@msu.edu‡ yuan@pa.msu.edu§ fyuan@lbl.gov

der (NNLO) in QCD interactions [20–22]. Recently,the transverse momentum effects from the initial statepartons are also discussed in the Z boson plus one jetproduction [23].

In this work, we focus on improving the prediction onthe kinematical distributions of the production of the Zboson plus one jet,

p+ p→ Z(PZ) + Jet(PJ) +X , (1)

where PZ and PJ are the momenta of Z boson and lead-ing jet, respectively. The transverse momentum resum-mation (q⊥ resummation) formalism is applied to sumover large logarithm ln(Q2/q2

⊥), with Q � q⊥, to all or-ders in the expansion of the strong interaction couplingat the NLO and next-leading logarithm (NLL) accuracy,where Q and q⊥ are the invariant mass and total trans-verse momentum of Z boson plus leading jet final statesystem, respectively. The q⊥ resummation technique isbased on the transverse momentum dependent (TMD)factorization formalism [24, 25], which has been widelydiscussed in the literature in the color singlet processes,such as Drell-Yan production [26]. Extending the q⊥ re-summation formalism to processes with more complexcolor structure have been discussed recently; e.g. heavyquark pair production [27–29]; processes involving mul-tijets in the final state [30–36]. Here we will use theTMD resummation formalism presented in Refs. [30, 32]to discuss the kinematical distributions of Z boson plusone jet. To properly describe the jet in the final state, weshould modify q⊥ resummation formalism to include thesoft gluon radiation from the final state; see a detaileddiscussion in Refs. [30–36]. In short, we should resumthe large logarithm ln(Q2/q2

⊥) when there is soft gluonradiation outside the observed final-state jet cone.

TMD Resummation. Our TMD resummation formula

arX

iv:1

810.

0380

4v3

[he

p-ph

] 2

5 Se

p 20

19

2

can be written as [31]:

d5σ

dyZdyJdP 2J⊥d

2~q⊥=∑ab[∫

d2~b⊥(2π)2

e−i~q⊥·~b⊥Wab→ZJ(x1, x2, b⊥) + Yab→ZJ

],

(2)

where yZ and yJ denote the rapidity of the Z boson and

the leading jet; PJ⊥(PZ⊥) and ~q⊥ = ~PZ⊥ + ~PJ⊥ are theleading jet (Z boson) transverse momentum and the im-balance transverse momentum of the Z boson and the jetsystem. The first term (Wab→ZJ) contains all order re-summation effect and the second term (Yab→ZJ) accountsfor the difference between the fixed order result and theso-called asymptotic result which is given by expandingthe resummation result to the same order in αs as thefixed order term. x1 and x2 are the momentum fractionsof the incoming hadrons carried by the two incoming par-tons,

x1,2 =

√m2Z + P 2

Z⊥e±yZ +

√P 2J⊥e

±yJ√S

, (3)

where mZ and S are the Z boson mass and squared col-lider energy, respectively. The all order resummation re-sult Wab→ZJ can be further written as,

Wab→ZJ (x1, x2, b)

= x1fa(x1, µF = b0/b⊥)x2fb(x2, µF = b0/b⊥)

×Hab→ZJ(s, µres, µR)e−SSud(s,µres,b⊥)e−FNP , (4)

where s = x1x2S, b0 = 2e−γE with γE being the Eu-ler constant, µres is the resummation scale to apply theTMD factorization in the resummation calculation. µres

is also the scale to define the TMDs in the Collins 2011scheme [37]. µR is the renormalization scale. fa,b(x, µF )are the PDFs for the incoming partons a and b, µF is fac-torization scale of the PDFs and b⊥ = b/

√1 + b2/b2max

with bmax = 1.5 GeV−1, which is introduced to factor outthe non-perturbative contribution eFNP , arising from thelarge b region (with b� b⊥) [38–41],

FNP (Q2,b) = g1b2 + g2 ln

Q

Q0ln

b

b⊥, (5)

where g1 = 0.21, g2 = 0.84 and Q20 = 2.4 GeV2 [41].

The Sudakov form factor can be expressed as,

SSud =

∫ µ2res

b20/b2⊥

dµ2

µ2

[ln

(s

µ2

)A+B1 +B2 +D ln

1

R2

],

(6)

where R denotes the jet cone size of the final state jet.The coefficients A, B1,2 and D can be expanded pertur-batively in αs, which is g2

s/(4π).

A/B1,2/D =

∞∑n=1

(αsπ

)nA(n)/B

(n)1,2 /D

(n). (7)

For qq̄ → Zg channel, we have

A(1) = CF , A(2) =1

2CFK, B

(1)1 = −3

2CF ,

B(1)2 = 0, D(1) =

1

2CA.

(8)

For gq → Zq channel, we have

A(1) =1

2(CF + CA), , A(2) =

1

2

CF + CA2

K,

B(1)1 = (−CAβ0 −

3

4CF ), B

(1)2 =

1

2(CF − CA) ln

(ut

),

D(1) =1

2CF , (9)

where CF =4

3, CA = 3 and K =

67

18− π2

6CA −

5

9Nf ;

β0 = (11 − 2/3Nf )/12 with Nf = 5 being the num-ber of effective light quarks. Here t = (Pa − PZ)2 andu = (Pa−PJ)2 with the incoming parton momentum Pa.They are the usual Mandelstam variables for the partonic2→ 2 process. The coefficients A and B1 come from theenergy evolution effect in the TMD PDFs [42], so thatthey only depend on the flavor of the incoming partonsand are independent of the scattering processes. The co-efficient B2 describes the soft gluon interaction betweeninitial and final states. The factor D quantifies the effectof soft gluon radiation which goes outside the jet cone,hence it depends on the jet cone size R. Furthermore, thenarrow jet approximation [43, 44] is applied to simplifythe calculation, and we only keep the term proportionalto ln(1/R2). In our numerical calculation, the A(2) termswill also be included in our analysis since it is associatedwith the incoming parton distribution and universal forall processes [45].

By applying the TMD factorization with Collins 2011scheme, we obtain the hard factor Hqq̄→Zg in Eq. (4), atthe one-loop order, as

3

H(1)qq̄→Zg = H

(0)qq̄→Zg

αs2π

{[−2β0 ln

(R2P 2

J

µ2res

)+

1

2ln2

(R2P 2

J

µ2res

)+ Li2

(m2Z

m2Z − t

)+ Li2

(m2Z

m2Z − u

)− ln

(µ2

res

m2Z

)ln

(sm2

Z

tu

)− 1

2ln2

(µ2

res

m2Z

)+

1

2ln2

(s

m2Z

)− 1

2ln2

(tu

m4Z

)+ ln

(t

m2Z

)ln

(u

m2Z

)+

1

2ln2

(m2Z − tm2Z

)+

1

2ln2

(m2Z − um2Z

)− 1

2ln2

(1

R2

)− 2π2

3+

67

9− 23Nf

54

]CA + 6β0 ln

µ2R

µ2res

+

[2 ln

(s

m2Z

)ln

(µ2

res

m2Z

)− ln2

(µ2

res

m2Z

)− 3 ln

(µ2

res

m2Z

)− ln2

(s

m2Z

)+ π2 − 8

]CF

}+ δH

(1)qq̄→Zg,

(10)

The leading order matrix element for qq̄ → Zg is,

H(0)qq̄→Zg =

8π

3αsCF

(g2V + g2

A

) [ut

+t

u+

2m2Z(m2

Z − u− t)tu

].

(11)

The vector and axial-vector gauge couplings between Z

boson and quarks are,

gV =gW

2 cos θW(τ q3 − 2Qq sin θ2

W ), gA =gW

2 cos θWτ q3 ,

(12)where gW and θW are the weak gauge coupling and weakmixing angle, respectively. τ3

q is the third component ofthe quark weak isospin and Qq is the electric charge of

quark. δH(1) represents terms which are not proportionalto H(0) and can be found in Ref. [7]. Similarly, for thesubprocess g + q → Z + q, we have

H(1)qg→Zq = H

(0)qg→Zq

αs2π

{[−3

2ln

(R2P 2

J

µ2res

)+

1

2ln2

(R2P 2

J

µ2res

)+ 2 ln

(u

m2Z

)ln

(µ2

res

m2Z

)− ln2

(µ2

res

m2Z

)− 3 ln

(µ2

res

m2Z

)− ln2

(u

m2Z

)− 1

2ln2

(1

R2

)− 2π2

3− 3

2

]CF + 6β0 ln

µ2R

µ2res

+

[−Li2

(m2Z

s

)+ Li2

(m2Z

m2Z − t

)− ln

(µ2

res

m2Z

)ln

(um2

Z

st

)− 1

2ln2

(µ2

res

m2Z

)− 1

2ln2

(st

m4Z

)+ ln

(s

m2Z

)ln

(t

m2Z

)− ln

(s

m2Z

)ln

(t

m2Z − s

)− 1

2ln2

(s

m2Z

)+

1

2ln2

(m2Z − tm2Z

)+

1

2ln2

(u

m2Z

)+π2

2

]CA

}+ δH

(1)qg→Zq,

(13)

where the leading order matrix element is,

H(0)qg→Zq = −παsCF

(g2V + g2

A

) [st

+t

s+

2m2Z(m2

Z − s− t)ts

].

(14)

We should note that the non-global logarithms (NGLs)could also contribute to this process. The NGLs arisefrom some special kinematics of two soft gluon radia-tions, in which the first one is radiated outside of thejet which subsequently radiates a second gluon into thejet [46–49]. Recently, the NGLs effects were studied inRef. [51] in the framework of soft-collinear effective the-ory and it shows that their contributions are negligiblewhen PJ⊥ > 30 GeV. Therefore we will not consider theNGLs in the following numerical calculations. The addi-

tional resummation effect of lnR is beyond the scope ofthis paper and has also been discussed in Ref. [51].

Z Boson Plus Jet Production at the LHC. We ap-ply the resummation formula of Eq. (2) to calculate thedifferential and total cross sections of the Z boson pro-duction associated with a high energy jet. The anti-ktjet algorithm with jet cone size R = 0.4 will be used todefine the observed jet as discussed in Refs. [32, 44].

Before we present our numeric results, we would like tocomment on the cross-check of our resummation method.We perform the fixed order expansion of the integral ofEq. (2) to obtain the total cross section, and compare itwith the fixed order prediction. The Y -term is vanishingwhen q⊥ goes to zero in the resummation framework, thusthe cross section in the small q⊥ region (from q⊥ = 0

4

to a small value q⊥,0, about 1 GeV) can be obtainedby integrating the distribution of the asymptotic partand the one-loop virtual diagram contribution. The crosssection in the large q⊥ region (q⊥ > q⊥,0) is infrared safeand can be numerically calculated directly. Thus, thetotal cross section can be written as [52],

σNLO =

∫ q2⊥,0

0

dq2⊥dσvirtual+realNLO

dq2⊥

+

∫ ∞q2⊥,0

dq2⊥dσrealNLO

dq2⊥

.

Numerically, we find that the above procedure reproducesthe NLO cross sections from MCFM [53] with slight dif-ference, ranging from 2% for R = 0.4 to 0.2% for R = 0.2.Clearly, this discrepancy arises from the narrow jet ap-proximation made in our derivations. Following the pro-cedure of Ref. [33], we parameterize this difference as

function of R: H(0)αs2π

(0.74R − 6.44R2) for the range of

0.2 < R < 0.6, which will be considered as part of ourNLO contribution H(1).

Recently, CMS collaboration has reported the mea-surement of the q⊥ spectrum of Z boson plus jet pro-duction at the 13 TeV LHC [5]. Since the experimentalmeasurement was done with certain kinematic cuts im-posed on the final state leptons, its result cannot be di-rectly compared to the current theory prediction which isfor an on-shell Z boson production associated with one ormore high-PJ⊥ jets. In order to compare to this data, weneed to estimate the effect of those kinematic cuts to ourtheory prediction. This estimation can be done by em-ploying a reweighting procedure based on the PYTHIA8simulations. For example, the differential cross sectionof the imbalance transverse momentum of the Z bosonand leading jet system (q⊥), after imposing the kinematiccuts on the decay leptons of the Z boson, can be writtenas

dσ

dq⊥

∣∣∣∣decay

=dσ

dq⊥

∣∣∣∣stable,Z

× κ(m`+`− , y`+`− , p`+`−

T ), (15)

where κ(m`+`− , p`+`−

T , y`+`−) is the reweighting factorwhich depends on lepton pair invariant mass (m`+`−),

transverse momentum (p`+`−

T ) and rapidity (y`+`−).dσ/dq⊥|stable,Z is the differential cross section with sta-ble Z boson production. Figure 1 shows the normal-

ized m`+`− , p`+`−

T and y`+`− distributions at the√S =

13 TeV, with |yJ | < 2.4 and PJ⊥ > 30 GeV, as predictedby the Monte Carlo event generator PYTHIA8 [54]. Theblue solid lines show the distributions after we imposethe following kinematic cuts on the leptons (labelled as‘with cut’) [5],

71 GeV < m`+`− < 111 GeV,

p`±

T > 20 GeV, and |η`± | < 2.4. (16)

The red dashed lines show the prediction of pp →γ∗/Z(→ `+`−) + jet without the above kinematic cutsimposed on the Z-decay leptons (labelled as ‘no cut’),while the black dotted lines show the prediction with

stable Z boson production (labelled as ‘stable Z’). Itis clear that the normalized distributions of m`+`− and

p`+`−

T are not sensitive to the imposed lepton kinematiccuts. On the contrary, the kinematic cuts on the decayleptons significantly modified the shape of the rapiditydistribution of the lepton pairs, cf. Fig. 1(c). Therefore,to a very good approximation, we can assume that thereweighting factor κ only depends on the value of y`+`− ,i.e.

κ(m`+`− , p`+`−

T , y`+`−) ' κ(y`+`−). (17)

The kinematic cuts imposed on the leptons, as inEq. (16), will constrain the allowed rapidity range of thelepton pair, and approximately |y`+`− | < 1.5. Figure1(d) shows the ratio of normalized rapidity distributionbetween with cut and stable Z boson prediction in Fig.1(c). It is clear that κ(y`+`−) does not strongly dependon y`+`− for |yZ | < 1.5. we could approximate a constantreweighting factor to describe the effect of the kinematiccuts on the Z-decay leptons; i.e.

κ(y`+`−) ' κ. (18)

Although κ is estimated based on LO prediction given bythe PYTHIA8 event generaotr, the theoretical uncertain-ties from higher order corrections are not significant [6].Therefore, under this approximation, we have

dσ

dq⊥

∣∣∣∣decay

' κ×(dσ

dq⊥

∣∣∣∣stable,Z

), for |y`+`− | < 1.5, (19)

with the kinematic cuts imposed in the CMS measure-ment [5]. For the normalized distribution, the κ depen-dence would be cancel out and yield the following rela-tions:

dσ

σdq⊥

∣∣∣∣decay

'(

dσ

σdq⊥

∣∣∣∣stable,Z

), for |y`+`− | < 1.5. (20)

This approximation is expected to hold well better thanthe theoretical uncertainty of the normalized q⊥ differen-tial cross section which is at the order of 10%, cf. Fig. 2.Hence, the small correction arising from taking into ac-count the full rapidity dependence of the re-weightingfactor can be ignored in this study.

We calculate the normalized q⊥ distribution of Z bo-son plus one jet production at the

√S = 13 TeV LHC

with CT14NNLO PDF [55], after imposing the kinematiccuts with |yJ | < 2.4 and PJ⊥ > 30 GeV, and the resultsare shown in Fig. 2(a). We fix the resummation scaleµres = PJ⊥, while the renormalization scale µR is takento be HT =

√m2Z + P 2

J⊥. The factorization scale µF ofthe Y -term is also taken to be HT . We estimate the scaleuncertainties in our calculation by simultaneously vary-ing the scales µR and µF by a factor of two around thecentral value HT with a correlated way. The blue andred bands represent the experimental uncertainty andscale uncertainty, respectively. In Fig. 2(b), we comparethe predictions from our resummation calculation to the

5

S =13 TeVHaL

pp®Γ*�ZH®{

+{

-L+jet

Èy J È<2.4

P J¦>30 GeV

with cut

no cut

70 80 90 100 1100.0

0.1

0.2

0.3

0.4

m{

+{

-@GeVD

dΣ

�dm

{+

{-

1�Σ

S =13 TeVHbL

pp®Γ*�ZH®{

+{

-L+jet

Èy J È<2.4

P J¦>30 GeV

With cut

no cut

stable Z

0 40 80 120 160 2000.00

0.02

0.04

0.06

0.08

0.10

pT

{+

{-

@GeVD

dΣ

�dp

T{+

{-

1�Σ

S =13 TeVHcLpp®Γ

*�ZH®{+

{-L+jet

Èy J È<2.4

P J¦>30 GeV

with cut

no cut

stable Z

-5 -3 -1 1 3 50.00

0.05

0.10

0.15

0.20

y{

+{

-

dΣ

�dy

{+

{-

1�Σ

S =13 TeV

HdL

pp®Γ*�ZH®{+

{-L+jet

Èy J È<2.4, P J¦>30 GeV

-1 0 10

1

2

3

4

y{+

{-

RHw

ith

cu

t�sta

ble

ZL

FIG. 1. Comparison of normalized differential distributions of the (a) invariant mass; (b) transverse momentum; and (c)

rapidity; of the lepton pairs predicted by PYTHIA8 for the Z-boson plus jet production at the√S = 13 TeV LHC with

|yJ | < 2.4 and PJ⊥ > 30 GeV. The blue sold lines show the distributions with the kinematic cuts imposed on the leptons,

as done in the CMS measurement [5], which are 71 GeV < m`+`− < 111 GeV, p`±

T > 20 GeV, and |η`± | < 2.4. The reddashed lines show the predictions without imposing the kinematic cuts on the decay leptons. The black dotted lines show

the predictions for a stable Z boson production, hence, p`+`−

T ≡ pZT and y`+`− ≡ yZ . (d) the ratio of normalized rapiditydistribution between with cut (blue solid line in (c)) and stable Z boson (black dashed line in (c)).

S =13 TeV

HaL

pp®Z+jet

Μres=P J¦, P J¦

>30 GeV

Èy J È<2.4, ÈyZ È<1.5

CMS

Res

0 20 40 60 800.0

0.1

0.2

0.3

0.4

q¦

@GeVD

dΣ

�dq

¦1

�Σ

S =13 TeVHbLpp®Z+jet

Μres=P J¦, P J¦

>30 GeV

Èy J È<2.4, ÈyZ È<1.5

CMS

Res

0 20 40 60 80

0.6

0.8

1.0

1.2

1.4

q¦

@GeVD

Re

s�d

ata

FIG. 2. (a) The normalized q⊥ distribution of the Z boson

plus one jet system, produced at the√S = 13 TeV LHC

with |yJ | < 2.4 and PJ⊥ > 30 GeV. The blue and red bandsrepresent the CMS experimental uncertainty [5] and the re-summation calculation (Res) scale uncertainty, respectively.(b) The ratio of resummation prediction to CMS data as afunction of q⊥.

CMS data by taking the ratio of their q⊥ differential dis-tributions. It is clear that our resummation calculationagree well with the experimental data. We also show thecomparision between resummation calculation and NLOprediction in Fig. 3(a). It is clear that there is a largedeviation between NLO and resummation calculation inthe small q⊥ region.

The azimuthal angle (φ) between the final state jetand Z boson measured in the laboratory frame is relatedto the q⊥ distribution, and is thus sensitive to the softgluon radiation. The advantage of studying the φ distri-bution is that it only depends on the moving directionsof the final state jet and Z boson. This observable wasmeasured by the CMS Collaboration at the 7 and 8 TeVLHC [3, 56]. In Figs. 4 and 5, we compare the normal-ized φ angle distribution at the 7 TeV and 8 TeV LHC,respectively. Similar to the q⊥ spectrum, the predictionsof our resummation calculation agree well with the CMS

S =13 TeV

HaL

pp®Z+jet

Μres=P J¦, P J¦

>30 GeV

Èy J È<2.4, ÈyZ È<1.5

Res

NLO

0 20 40 60 800

1

2

3

4

5

q¦

@GeVD

dΣ

�dq

¦@n

bD S =8 TeV

HbL

Μres=P J¦, P J¦

>30 GeV

Èy J È<2.4, ÈyZ È<1.5

Res

NLO

2.4 2.6 2.8 3.00

10

20

30

40

Φ @radD

dΣ

�dΦ

@pb

�rad

D

FIG. 3. The q⊥ (a) and φ (b) distributions from resummationcalculation (blue band) and NLO prediction (red band) at the√S = 13 TeV and

√S = 8 TeV LHC, with |yJ | < 2.4 and

PJ⊥ > 30 GeV, respectively.

data. The comparision between NLO and resummationcalculation is shown in Fig. 3(b).

Summary. In summary, we have applied the TMD re-summation formalism to study the production of the Zboson associated with a high energy jet at the LHC,where large logarithms of ln(Q2/q2

⊥) were resumed to allorders at the NLL accuracy. We also calculate the NLOtotal cross section based on the resummation frameworkand the result is slightly different from the MCFM pre-diction due to the usage of narrow jet approximation inour resummation calculation. To ensure the correct NLOtotal cross section, we have added an additional term pro-portional to H(0) to account for the above difference inour resummation calculation. To compare the predictionof our resummation calculation (for an on-shell Z bo-son) to the CMS experimental data (with kinematic cutsimposed on Z-decay leptons), we approximate the effectof imposing kinematic cuts on the Z-decay leptons byemploying a reweighting procedure based on the resultof PYTHIA8 prediction. It shows that we could use aconstant reweighting factor to describe the effects of the

6

S =7 TeV

HaL

pp®Z+jet

Μres=P J¦, P J¦

>50 GeV

Èy J È<2.5, ÈyZ È<1.5

CMS

Res

2.4 2.6 2.8 3.00

1

2

3

4

Φ @radD

dΣ

�dΦ

1�Σ

@1�r

ad

DS =7 TeVHbL

pp®Z+jet

Μres=P J¦, P J¦

>50 GeV

Èy J È<2.5, ÈyZ È<1.5

CMS

Res

2.4 2.6 2.8 3.0

0.6

0.8

1.0

1.2

1.4

Φ @radD

Re

s�d

ata

FIG. 4. The normalized distribution of φ, the azimuthal anglebetween the final state jet and Z boson measured in the labo-ratory frame, for pp→ Z+jet production at the

√S = 7 TeV

LHC with |yJ | < 2.5, |yZ | < 1.5 and PJ⊥ > 50 GeV. Theblue and red bands represent the CMS experimental uncer-tainty [56] and the resummation calculation (Res) scale un-certainty, respectively.

S =8 TeV

HaL

pp®Z+jet

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FIG. 5. Same as Fig. 4, but for the 8 TeV LHC [3].

kinematic cuts imposed on the Z-decay leptons. A de-tailed comparison between our resummation calculationand the CMS data is also discussed. We find that our re-summation calculation can describe well the CMS data,both in the distributions of the imbalance transverse mo-mentum (q⊥) and the azimuthal angle (φ) correlation ofthe Z boson and jet system, for pp→ Z+ jet productionat the LHC

Acknowledgment: This work is partially supported bythe U.S. Department of Energy, Office of Science, Officeof Nuclear Physics, under contract number DE-AC02-05CH11231, and by the U.S. National Science Founda-tion under Grant No. PHY-1719914. C.-P. Yuan is alsograteful for the support from the Wu-Ki Tung endowedchair in particle physics.

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