IR-improved DGLAP-CS parton shower effects in W + jets in pp collisions at $\sqrt{s}=7$ TeV

We use HERWIRI1.031, a new Monte Carlo (MC) for hadron-hadron scattering at high energies, to study the phenomenological effects of our approach of exact amplitude-based resummation in precision QCD calculations. W + jet(s) events with exact NLO QCD corrections are generated in the MG5_aMC@NLO framework and showered by both HERWIRI1.031 and HERWIG6.5 with PTRMS = 0 and PTRMS = 2.2 GeV/c, respectively. Here, PTRMS is the rms value of the intrinsic Gaussian transverse momentum distribution for the partons inside the proton. The differential cross sections for many observables are presented such as the jet rapidities and the jet transverse momenta as well as other event observables such as the scalar sums of transverse momenta of the jets, the missing transverse energy of the jets and the dijets' observables. Finally, we compare our results with the ATLAS and CMS measurements of the W production cross sections in association with jets.

the Herwiri1.031 [6] IR-improved shower to be compared with the standard unimproved Herwig6.5 [8] shower in that framework. In this way, we realize exact NLO matrix element matched parton showers with and without IR-improvement. We compare with the data from ATLAS and CMS at 7 TeV to make contact with observations. The paper is organized as follows. In the next section we give a brief review of exact QED ⊗ QCD resummation theory. In Section 3 we describe our event generation, analysis and cuts. In Section 4 we compare our predictions with the ATLAS 7 TeV data. In Section 5 we compare our predictions with the CMS 7 TeV data. Section 6 contains our concluding remarks.

II. EXTENSION OF YFS THEORY TO QED ⊗ QCD
We start with a prototypical process pp → W ± + n(γ) + m(g) + X → l ± + ν l ± + n (γ) + m(g) + X , where l = {e, µ}, ν l + = ν l , and ν l − =ν l . The new QED ⊗ QCD YFS extension is obtained by simultaneously resumming the large IR terms in QCD and the IR dominant terms in QED. One can prove that the exponentiated cross section is given by [9] dσ exp = with n(γ) hard photons and m(g) hard gluons, whereβ n,m (k 1 , ..., k n ; k 1 , ..., k m ) are the YFS residuals which are free of all infrared divergences to all orders in α s and α. The infrared functions are given by SUM IR (QCED) = 2α s ReB nls QCED + 2α sB nls QCED (K max ), and the functions SUM IR (QCED), D QCED are determined form their QCD analougs SUM IR (QCD), D QCD via the following substitutions In Eq (5), the superscript nls asserts that the infrared functions B QCD , B QED ,B QCD ,B QED andS QCD are DGLAP-CS synthesized. These infrared functions have been introduced in Ref. [10]. The QCD exponentiation of the master formula in Eq (1) leads to a new set of IR-improved splitting functions listed below where Finally, for precision LHC theory, the famous factorization theorem [11] is written in the following form where the primed quantities are associated with the kernels and cross sections derived in Eqs (6) and (1) respectively. The implementation of the new IR-improved kernels in the HERWIG6.5 [8] environment leads to a new MC, HERWIRI1.031, as described in Ref. [12].
In what follows, we present results using both the original Herwig6.5 and the new IRimproved Herwiri1.031

III. EVENT GENERATION, ANALYSIS AND CUTS
The generators for W + jet events are MADGRAPH5 aMC@NLO [7] interfaced with HERWIG6.521 and HERWIRI1.031, which use with exact next-to-leading-order (NLO) matrix element calculations matched to the respective parton shower. The number of events generated for the W, W + 1 jet, W + 2 jets, and W + 3 jets processes are 10 7 , 10 6 , 10 5 , and 10 5 , respectively. These events are showered by MADGRAPH5 aMC@NLO/HERWIRI1.031 (PTRMS = 0) and MADGRAPH5 aMC@NLO/HERWIG6.521 (PTRMS = 2.2 GeV). 1 During the analysis, jets were reconstructed using the anti-k t algorithm with FastJet [13] and the cuts in Tables I and II were imposed for the ATLAS and CMS results, respectively. 1 We will see later that HERWIRI gives either a better fit to the data or an acceptable fit without this extra intrinsic Gaussian kick.  The transverse mass, m T , is defined as m T = 2P l T P ν l T (1 − cos ∆φ) where ∆φ is the difference in the azimuthal angle between the direction of the lepton momentum and the associated neutrino, ν l , which can be written as Rapidity is defined as where E denotes the energy of the particle and p z is the longitudinal component of the momentum. Finally, the jet isolation, ∆R, which is a Lorentz invariant quantity for massless particles, is defined as ∆R(l, jet) = ∆φ 2 (l, jet) + ∆η 2 (l, jet), where where θ is the angle between the respective particle three-momentum P and the positive direction of the beam axis. The E miss T is calculated as the negative vector sum of the transverse momenta of calibrated leptons, photons and jets and additional low-energy deposits in the calorimeter.

IV. RESULTS (ATLAS COLLABORATION)
In this section, the measured W(→ l + ν l ) + jets fiducial cross sections [14] are shown and compared to the predictions of MADGRAPH5 aMC@NLO/HERWIRI1.031 and MAD-GRAPH5 aMC@NLO/HERWIG6.521. Each distribution is combined separately by minimizing a χ 2 function. The factors applied to the theory predictions are summarized in Appendix A.

A. Transverse Momentum Distributions
The differential cross sections as a function of the leading jet transverse momentum are shown in Figure. 1 and Figure. 2 for the W + ≥1 jet and W + 1 jet cases, respectively. In both cases, there is agreement between the data and predictions provided by HERWIRI and HERWIG in the soft regime.
In Figure.      The differential cross sections for the production of W + ≥2 jets as a function of the leading jet P T and the second leading jet P T are shown in Figure. 3 and Figure. 4, respectively.
HERWIRI and HERWIG generally describe the data well for P T < 200 GeV. In Figure. 3,   The differential cross sections for the production of W + ≥3 jets as a function of the leading jet P T and the third leading jet P T are shown in Figure In general, one could conclude that the predictions provided by HERWIRI give as good a fit or a better fit to the data for soft P T without the need of an 'ad hoc' intrinsic Gaussian rms transverse momentum of 2.2 GeV as needed by HERWIG.

B. Rapidity Distributions
The differential cross sections for the production of W + ≥1 jet as a function of the leading jet Y j are shown in Figure. 7. The predictions provided by HERWIRI and HERWIG are generally in agreement with the data, although in three cases HERWIRI predictions overlap with the data while the HERWIG predictions either underestimate or overestimate the data.
We clearly conclude that HERWIRI gives a better fit to the data with The differential cross sections for the production of W + ≥2 jets as a function of the second leading jet Y j are shown in Figure. 8. The results provided by HERWIRI and HER-WIG overlap with the data in almost all cases. In two cases, the HERWIRI predictions overlap with the data and in two cases the HERWIG results overlap with the data while HERWIRI predictions either underestimate or overestimate the data:    The differential cross sections for the production of W + ≥ 3 jets as a function of the third leading jet Y j are shown in Figure. 9. For Y j < 3.6, with the exception of one case in which only the HERWIG prediction overlaps with the error bars on the data, HERWIRI and HERWIG predictions are in agreement with the data. For Y j > 3.6, in one case HERWIRI overlaps with the error bars on the data while HERWIG overestimates the data, and in the other case HERWIG overlaps with the error bars on the data while HERWIRI underestimates the data:

C. Dijet Angular Variables, Invariant Mass, Separation
In this subsection the differential cross sections are shown as functions of the difference in azimuthal angle (∆φ j 1 ,j 2 ), the difference in the rapidity (∆Y j 1 ,j 2 ), the angular separation (∆R j 1 ,j 2 ) and the dijet invariant mass (m j 1 ,j 2 ) in comparison to the data. We define the aforementioned variables as follows We note that in Eq. (4.6), η j 1 ,j 2 is the difference in rapidity of the first and second leading jets. The ith jet is defined as Figure. 10: Cross section for the production of W + jets as a function of the dijet invariant mass m j 1 ,j 2 between the two leading jets in N jet ≥ 2. The data are compared to predictions from MADGRAPH5 aMC@NLO/HERWIRI1.031 and MADGRAPH5 aMC@NLO/HERWIG6.521.
The differential cross sections for the production of W + ≥2 jets as a function of the dijet invariant mass between the two leading jets are shown in Figure. 10 The differential cross sections for the production of W + ≥2 jets as a function of the difference in the rapidity between the two leading jets are shown in Figure. 11. For ∆Y j 1 j 2 < 3 the predictions provided by HERWIRI give a better fit to the data. For 3 < ∆Y j 1 j 2 < 4, HERWIG results provide a better description of the data:  The differential cross sections for the production of W + ≥2 jets as a function of the angular separation between the two leading jets are shown in Figure. 12. For ∆R j 1 ,j 2 > 3, the cross sections are fairly well modeled by the predictions of HERWIRI and HERWIG. For ∆R j 1 ,j 2 < 3, in at least two cases the prediction provided by either of them are outside of the error bars on the data; in most cases they both give a satisfactory prediction relative to the data: The differential cross sections for the production of W + ≥2 jets as a function of the azimuthal angle between the two leading jets are shown in Figure. 13. For ∆φ j 1 ,j 2 < 0.4, 1 < ∆φ j 1 ,j 2 < 1.4, and ∆φ j 1 ,j 2 > 2.2, the predicted cross sections by HERWIRI and HERWIG are within the error bars on the data: while both predictions give acceptable fits to the data, the HERWIG fit is the better one. Figure. 12: Cross section for the production of W + jets as a function of the angular separation between the two leading jets for N jet ≥ 2. The data are compared to predictions from MAD-GRAPH5 aMC@NLO/HERWIRI1.031 and MADGRAPH5 aMC@NLO/HERWIG6.521. Figure. 13: Cross section for the production of W + jets as a function of the difference in the azimuthal angle between the two leading jets in N jet ≥ 2. The data are compared to predictions from

D. Scalar Sum H T
In this subsection we will study the W + jets cross sections as a function of H T , the summed scalar P T of all identified objects in the final state. For example, for a prototypical process pp → l + ν l + j 1 + j 2 , we define H T as follows where l = e, µ.
The differential cross sections as a function of H T are shown in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, and Figure 19 respectively. We will study the W + jets cross sections as a function of H T for low H T . We will see in some cases HERWIRI predictions are in agreement with the data and in some cases HERWIG predictions give a better fit to the data. In general, a better agreement is provided for the lower jet multiplicities, e.g.
The differential cross sections for the production of W + ≥1 jet as a function of the scalar sum H T are shown in Figure. 14. For H T < 300 GeV, HERWIRI predictions are in better agreement with data where: The differential cross sections for the production of W + 1 jet as a function of the scalar sum H T are shown in Figure.    The differential cross sections for the production of W + 2 jets as a function of the scalar sum H T are shown in Figure. 17. HERWIRI and HERWIG seem to be unable to provide a good fit for the data at H T < 190 GeV where they underestimate the data; In the H T < 250 GeV range, HERWIG predictions are in better agreement with the data, where   At scalar sum values around 170 < H T < 250 GeV, HERWIRI and HERWIG predictions overlap fairly well with the data. In general, we conclude that the discrepancy of the predictions provided by HERWIRI is less than that of HERWIG.
The differential cross sections for the production of W + ≥3 jets as a function of the scalar sum H T are shown in Figure.  The differential cross sections for the production of W + 3 jets as a function of the scalar sum H T are shown in Figure. 19. HERWIG gives a better fit to the data for H T < 250,

E. Scalar Sum S T
In this subsection, we study the behavior of W + jets cross sections as a function of the scalar sum S T , where S T is defined as the summed scalar P T of all the jets in the event: where |P T (i)| is the transverse momentum of the ith jet and Njet is the maximum number of jets in each event. The differential cross sections as a function of S T are shown in Figure. 20, Figure. 21, Figure. 22, Figure. 23, and Figure. 24 respectively. We will study the W + jets cross sections as a function of S T for low S T . We will see in some cases HERWIRI predictions are in agreement with the data and in some cases HERWIG predictions give a better fit to the data. In general, a better agreement is provided for the lower jet multiplicities, e.g.
W + 1 jet and W + ≥ 1 jet.  gives either a better fit to the data or less discrepancy in comparison with HERWIG.
The differential cross sections for the production of W + ≥3 jets as a function of the scalar sum S T are shown in Figure.  It is clear in some cases HERWIRI predictions are in agreement with the data and in some cases HERWIG predictions give a better fit to the data. In general, a better agreement is provided for the lower jet multiplicities, e.g. W + 1 jet and W + ≥ 1 jet.

F. Cross Sections
The cross sections for W → l + ν l production as functions of the inclusive and exclusive jet multiplicity are shown in Figure. 25 and Figure. 26. Figure. 25 shows the cross sections for the production of W + jet as a function of the inclusive jet multiplicity. A good fit is provided by HERWIRI and HERWIG for N jet ≥ 1, for N jet ≥ 2 and for N jet ≥ 3, where the HERWIRI prediction is just at edge of the lower error bar on the data. For the exclusive case in Fig. 26, similar comments apply except that for the N jet = 3 case the HERWIRI prediction is about 2 σ below the data.

V. RESULTS (CMS COLLABORATION)
In this Section the measured W(→ µ + ν µ ) + jets fiducial cross sections [15] are shown and compared to the predictions of MADGRAPH5 aMC@NLO/HERWIRI1.031 and MAD-GRAPH5 aMC@NLO/HERWIG6.521,which are hereafter oftentimes referred to as HER-WIRI and HERWIG, respectively. Each distribution is combined separately by minimizing a χ 2 function. The factors applied to the theory predictions are summarized in Appendix B.

A. Transverse Momentum Distributions P T
The differential cross sections in jet P T for inclusive jet multiplicities from 1 to 3 are shown in Figure. 27, Figure.

B. The Scalar Sum of Jet Transverse Momenta H T
In this subsection, the differential cross sections are shown as function of H T for inclusive jet multiplicities 1-3. The scalar sum H T is defined as for each event.
The differential cross sections as a function of H T for inclusive jet multiplicities 1-3 are shown in Figure. 30, Figure. 31, and Figure. 32. In Figure.  gives a better fit to the data while in Figure.

C. Pseudorapidity Distributions |η(j)|
In this section, the differential cross sections are shown as functions of pseudorapidities of the three leading jets. The pseudorapidity, which was defined in Eq. (12), can be written where P L where is the component of the momentum along the beam axis.
The problem with rapidity is that it can be hard to measure for highly relativistic particles.
We need the total momentum vector of a particle, especially at high values of the rapidity where the z component of the momentum is large, and the beam pipe can be in the way of measuring it precisely.
However, there is a way of defining a quantity that is almost the same thing as the rapidity which is much easier to measure than y for highly energetic particles. This leads to the concept of the pseudorapidity η, wherein we see from Eq.(22) that the magnitude of the momentum cancels out of the ratio in the arguments of the logarithm and the arctanh in the equation.
Hadron colliders measure physical momenta in terms of transverse momentum, P T , polar angle in the transverse plane, φ, and pseudorapidity. To obtain Cartesian momenta (P x , P y , P z ), (with the z-axis defined as the beam axis), the following conversions are used: (23) Figure. 33: Cross section for the production of W + jets as a function of |η(j 1 )| for N jet ≥ In Figure. 33 the cross section is shown as a function of |η(j 1 )|, the leading jet pseudorapidity. The predictions provided by HERWIRI are in better agreement with the data, with

D. Azimuthal Angular Distribution Between the Muon and the Leading Jet
The differential cross sections are shown as functions of the azimuthal angle between the muon and the first three leading jets for inclusive jet multiplicities 1-3. The azimuthal angle between the muon and the leading jet is defined as cos(∆Φ(µ, j 1 )) = P x (µ)P x (j 1 ) + P y (µ)P y (j 1 ) Figure. 36: Cross section for the production of W + jets as a function of the azimuthal angle between the muon and the leading jet ∆Φ(µ, j 1 ) for N jet ≥ 1. The data are compared to predictions from MADGRAPH5 aMC@NLO/HERWIRI1.031 and MADGRAPH5 aMC@NLO/HERWIG6.521.
with      µ µ = (E µ , P x (µ), P y (µ), P L (µ)), The differential cross sections as functions of the azimuthal angle between the muon and the first three leading jets are shown in Figure. 36, Figure. 37, and Figure. 38 for inclusive jet multiplicities 1-3, respectively.
In Figure. 36 and Figure. 38, the data are better modeled by the predictions provided by HERWIRI as expected as well as Figure. 37 shows that the HERWIG predictions give a better fit to the data. In Figure.

E. Cross Sections
The measured W(→ µ + ν µ ) + jets fiducial cross sections are shown in Figure. 39 and Figure. 40 and compared to the predictions of MADGRAPH5 aMC@NLO/ HERWIRI1.031 and MADGRAPH5 aMC@NLO/HERWIG6.521. Figure. 39 shows the differential cross sections for the inclusive jet multiplicities 1-3. HERWIRI gives a better fit to the data. Figure. 40 shows the differential cross sections for the exclusive jet multiplicities 1-3. The cross sections provided by HERWIG give a better fit to the data. In Figure.

VI. SUMMARY
The realization of the IR-improved DGLAP-CS theory, when used in the MAD-GRAPH5 aMC@NLO/HERWIRI1.031 O(α) ME-matched parton shower framework, provides us with the opportunity to explain, in the soft regime, the differential cross sections for a W boson produced in association with jets in pp collisions in the recent LHC data from ATLAS and CMS, without the need of an unexpectedly hard intrinsic Gaussian distribution with an rms value of PTRMS = 2.2 GeV in parton's wave function. In our view, this can be interpreted as providing a rigorous basis for the phenomenological correctness of such unexpectedly hard distributions insofar as describing these data using the usual unimproved DGLAP-CS showers is concerned.