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

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 logarithms introduced by the small total transverse momentum of the $Z$ boson plus jet final state system, are resummed to all orders in the expansion of the strong interaction coupling at the accuracy of Next-to-Leading Logarithm(NLL). We also compare our result with the prediction of the Monte Carlo event generator PYTHIA8 and found around 10\% difference in the distribution of the total transverse momentum and the azimuthal angle correlations of the final state $Z$ boson and jet system.

Introduction. The Z boson and jets associated production at Large Hadron Collider (LHC) plays an important role in our knowledge of the Standard Model (SM) and beyond. The clean and readily identifiable signature and large production rate of this process provide an opportunity to precisely measure the electroweak parameters, constrain the parton distribution functions (PDFs) and also probe the strong coupling constant α s . In particular, it is a prominent background in searches for SM processes and physics beyond the SM at the TeV scale [1]. Therefore, a precise study of both the inclusive and differential measurements of Z boson plus jets production is vital to test the SM and search new physics (NP).
Currently, both the ATLAS and CMS collaborations have report the measurements of Z boson production associated with zero, one and two jets [2][3][4][5]. Although the experimental measurements show very good agreement with theoretical predictions [6], a better theoretical calculation is still needed to reduce the theoretical uncertainties. Both the fixed-order and resummation techniques could be used to improve the theoretical predictions. Perturbative QCD correction to the Z boson plus multijets production at the next-to-leading order (NLO) are widely discussed in literatures [7][8][9][10][11][12][13][14]. The NLO effects from electroweak correction to Z boson plus multijets are also discussed in Refs. [15][16][17][18]. Beyond the NLO QCD calculation, the leading threshold logarithms have been included in Ref. [19]. The accuracy to the Z boson plus one jet production has reached to the next-to-nextto-leading order (NNLO) in QCD interactions [20][21][22]. Recently, the transverse momenta effects from the initial state partons are also discussed in the Z boson plue one jet production [23].
In this work, we focus on improving the prediction on the kinematical distributions of the inclusive production * pengsun@msu.edu † yanbin1@msu.edu ‡ yuan@pa.msu.edu § fyuan@lbl.gov of the Z boson plus one jet, where P Z and P J are the momenta of Z boson and jet, respectively. The transverse momentum resummation (q ⊥ resummation) formalism is applied to sum over large logarithm ln(Q 2 /q 2 ⊥ ), with Q q ⊥ , to all orders in the expansion of the strong interaction coupling at the NLO and next-leading logarithm (NLL) accuracy, where Q and q ⊥ are the invariant mass and total transverse momentum of Z boson plus one jet final state system, respectively. The q ⊥ resummation technique is based on the transverse momentum dependent (TMD) factorization formalism [24,25], which has been widely discussed in the literature in the color singlet processes, such as Drell-Yan production [26]. Extending the q ⊥ resummation formalism to processes with more complex color structure have been discussed recently; e.g. heavy quark pair production [27][28][29]; processes involving multijets in the final state [30][31][32][33][34][35][36]. Here we will use the TMD resummation formalsim presented in Refs. [30,32] to discuss the kinematical distributions of Z boson plus one jet. To properly describe the jet in the final state, we should modify q ⊥ resummation formalism to include the soft gluon radiation from the final state; see a detailed discussion in Refs. [30][31][32][33][34][35][36]. In short, we should resum the large logarithm ln(Q 2 /q 2 ⊥ ) when the soft gluon radiation outside the observed final-state jet cone. TMD Resummation. Our TMD resummation formula can be written as [31]: where y Z and y J denote the rapidity of the Z boson and the jet; P J⊥ (P Z⊥ ) and q ⊥ = P Z⊥ + P J⊥ are the jet (Z arXiv:1810.03804v1 [hep-ph] 9 Oct 2018 boson) transverse momentum and the imbalance transverse momentum of the Z boson and the jet system. The first term (W ab→ZJ ) contains all order resummation effect and the second term (Y ab→ZJ ) accounts for the difference between the fixed order result and the so-called asymptotic result which is given by expanding the resummation result to the same order in α s as the fixed order term. x 1 and x 2 are the momentum fractions of the incoming hadrons carried by the two incoming partons, where m Z and S are the Z boson mass and squared collider energy, respectively. The all order resummation result W ab→ZJ can be further written as, where s = x 1 x 2 S, b 0 = 2e −γ E with γ E being the Euler constant, µ res is the resummation scale to apply the TMD factorization in the resummation calculation. µ res is also the scale to define the TMDs in the Collins 2011 scheme [37]. f a,b (x, µ) are the PDFs for the incoming partons a and b, µ is factorization scale of the PDFs which is introduced to factor out the non-perturbative contribution e F N P , arising from the large b region (with b b ⊥ ) [38][39][40][41], where g 1 = 0.21, g 2 = 0.84 and Q 2 0 = 2.4 GeV 2 [41].
The Sudakov form factor can be expressed as, where R denotes the jet cone size of the final state jet. The coefficients A, B and D can be expanded perturbatively in α s , which is g 2 s /(4π). For qq → Zg channel, at one-loop order, we have For gq → Zq channel, we have where C F = 4 3 and C A = 3. Here t = (P a − P Z ) 2 and u = (P a − P J ) 2 with the incoming parton momentum P a , are the usual Mandelstam variables for the partonic 2 → 2 process. The coefficients A and B 1 come from the energy evolution effect in the TMD PDFs [42], so that they only depend on the flavor of the incoming partons and are independent of the scattering processes. The coefficient B 2 describes the soft gluon interaction between initial and final states. The factor D quantifies the effect of soft gluon radiation which goes outside the jet cone, hence it depends on the jet cone size R. Furthermore, the narrow jet approximation [43,44] is applied to simplify the calculation, and we only keep the term proportional to ln(1/R 2 ). In our numerical calculation, the A (2) terms will also be included in our analysis since it is associated with the incoming parton distribution and universal for all processes [45]. By applying the TMD factorization with Collins 2011 scheme, we obtain the hard factor H qq→Zg in Eq. (4), at the one-loop order, as where β 0 = (11 − 2/3N f )/12 with N f = 5 being the number of effective light quarks. The leading order ma-trix element for qq → Zg is, The vector and axial-vector gauge couplings between Z boson and quarks are, where g W and θ W are the weak gauge coupling and weak mixing angle, respectively. τ 3 q is the third component of the quark weak isospin and Q q is the electric charge of quark. δH (1) represents terms which are not proportional to H (0) and can be found in Ref. [7]. µ R denotes the renormalization scale. Similarly, for the subprocess g + q → Z + q, we have where the leading order matrix element is, ts .
(13) We should note that the non-global logarithms (NGLs) could also contribute to this process. The NGLs arise from some special kinematics of two soft gluon radiations, in which the first one is radiated outside of the jet which subsequently radiates a second gluon into the jet [46][47][48][49]. Numerically, the NGLs are negligible in this process since it starts at O(α 2 s ) [50]. Therefore we will ignore their contributions in the following phenomenology discussion. Z Boson Plus Jet Production at the LHC. We apply the resummation formula of Eq. (2) to calculate the differential and total cross sections of the Z boson production associated with a high energy jet. The anti-k t jet algorithm with jet cone size R = 0.4 will be used to define the observed jet as discussed in Refs. [32,44].
Before we present our numeric results, we would like to comment on the cross-check of our resummation method. We perform the fixed order expansion of the integral of Eq.
(2) to obtain the total cross section, and compare it with the fixed order prediction. The Y -term is vanishing when q ⊥ goes to zero in the resummation framework, thus the cross section in the small q ⊥ region (from q ⊥ = 0 to a small value q ⊥,0 , about 1 GeV) can be obtained by integrating the distribution of the asymptotic part and the one-loop virtual diagram contribution. The cross section in the large q ⊥ region (q ⊥ > q ⊥,0 ) is infrared safe and can be numerically calculated directly. Thus, the total cross section can be written as [51], Numerically, we find that the above reproduce the NLO cross sections from MCFM [52] with slight difference, ranging from 2% for R = 0.4 to 0.2% for R = 0.2. Clearly, this discrepancy arises from the narrow jet approximation we made in our derivations. Following the procedure of Ref. [33], we parameterrize this difference as function of R: H (0) α s 2π (0.74R − 6.44R 2 ) for the range of 0.2 < R < 0.6, which will be considered as part of our NLO contribution H (1) . We calculate various differential cross section of Z boson plus one jet production at the √ S = 13 TeV LHC with CT14NNLO PDF [53], and results are shown in Fig. 1. Both the resummation scale (µ res ) and renormalization scale (µ R ) are taken to be H T = m 2 Z + P 2 J⊥ + P J⊥ . We also impose the following kinematic cuts with |y J | < 2.4 and P J⊥ > 30 GeV. Clearly, the NLO prediction (red dotted line) for the q ⊥ distribution is not reliable when q ⊥ is small. The resummation calculation predicts a well behavior q ⊥ distribution in the small q ⊥ region since the large logarithms have been properly resummed. The peak of q ⊥ spectrum is around 10 GeV, which is dominated by quark gluon initial state subprocess; see Fig. 2.
In Fig. 3, we estimate the scale uncertainties from resummation calculation by varying the scale µ Res = µ R by a factor of two around the central value H T . As a comparison, we also show the prediction from the parton shower event generator PYTHIA 8 [54] in Fig. 3 (red  solid line). A clear difference is found in the shape of the q ⊥ distribution.
The azimuthal angle (φ) between the final state jet and Z boson measured in the laboratory frame is related to the q ⊥ distribution, which is also sensitive to the soft gluon radiation. The advantage of studying the φ distribution is that it only depends on the moving directions of the final state jet and Z boson. In Fig. 4, we show the normalized φ angle distribution. Similar to the q ⊥ spectrum, a difference is found between resummation calcula-  tion (blue solid line) and PYTHIA prediction (red dashed line). To quantify the difference between our resummation calculation and PYTHIA prediction, we compare the predicted kinematic acceptance after we imposing the φ value cut in Table I. It is clear that the difference between resumation calculation and PYTHIA prediction is around 10%.
Since Z boson plus one jet production is a prominent background in Higgs plus jet production, it would be useful to compare the φ angle distribution between these two production processes; see Fig. 5. It shows the φ angle distribution is similar for these production processes since the difference of Sudakov factor in those two processes is small. The very small difference in the final results between these two processes comes from a combined effects of the resummation W terms and Y terms. Summary. In summary, we have applied the TMD resummation formalism to study the production of the Z boson associated with a high energy jet at the LHC, where large logarithms of ln(Q 2 /q 2 ⊥ ) was resumed to all orders at the NLL accuracy. We also calculate the NLO total cross section based on the resummation framework and the result is slightly different from the MCFM prediction due to the usage of narrow jet approximation in our resummation calculation. To ensure the correct NLO total cross section, we have added an additional term proportional to H (0) to account for the above difference in our resummation calculation. A detailed comparison between our theory calculation and PYTHIA prediction is also discussed. We find a clear difference between the PYTHIA and our resummation predictions in the distributions of the total transverse momentum (q ⊥ ) and the azimuthal angle (φ) correlations of the Z boson and jet system. Finally, we present a comparison of the azimuthal angular φ distributions of Z+jet and Higgs +jet processes and find little difference in the normalized distributions between them.
This work is partially supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-AC02-05CH11231, and by the U.S. National Science Foundation under Grant No. PHY-1719914. C.-P. Yuan is also grateful for the support from the Wu-Ki Tung endowed chair in particle physics.