Search

We study the differential cross section of the single inclusive $e^+e^-$ annihilation to the hadrons via $\gamma$-production, in the different ${k_{t}}$-factorization frameworks. In order to take into account the transverse momenta of the incoming partons, for the first time, we apply the Kimber et al (KMR) method to calculate the unintegrated parton fragmentation functions (UFFs) from the ordinary integrated one, i.e., the parton fragmentation functions (FFs), which satisfy the similar DGLAP evolution equations, such as those of parton distribution functions (PDFs). Also, by utilizing the different angular ordering constraints the results corresponding to the Martin et al (MRW) in the leading order (LO) and the next-to-leading-order (NLO) are obtained. The LO sets of DSS library for the input FFs is used. The numerical results are compared with the experimental data in the different energies which are reported by the different collaborations, such as TASSO, AMY, MARK II, CELLO, DELPHI, SLD, ALEPH and Belle with the other QCD+fragmentation models such as Pythia 6.4 and 8.2 parton showers. The behaviors of the normalized differential cross sections and the multiplicity versus the "transverse momentum" ($p_\bot$) are discussed. The final results demonstrate that the KMR and MRW UFFs give a good description of data and there is not much significant difference between the above three schemes. On the other hand, our results become closer to the data for the lower values of $p_\bot$ and the higher values of center of mass energies., Comment: Phys.Rev. D to accepted for publication

The quark exchange model (QEM) is reformulated for the $A=12$ and $14$ systems to obtain the constituent quark distributions of $^{12}C$ and $^{14}N$ nuclei, respectively. Afterwards, the different types of the point-like parton distribution functions (PDFs), i.e., the valence quarks, the sea quarks and the gluons, are extracted from the constituent quark model (CQM), at the hadronic scale $Q_0^2=0.34$ $GeV^2$ to generate the constituent quark exchange model (CQEM = QEM $\oplus$ CQM) PDFs. From the resulted PDFs, the structure functions and the European Muon Collaboration (EMC) ratios of the above-mentioned nuclei at the appropriate hadronic scale are calculated. To make our results more comparable with the available experimental data, we evolve the PDFs, by using the standard DGALP evolution equations, to the higher hard scales. Subsequently, the EMC ratios of $^{12}C$ and $^{14}N$ nuclei at the some higher energy scales, corresponding to the scales of available data, are calculated, at the leading (LO) and the next-to-leading (NLO) orders of pQCD. By doing so, it is observed that the EMC ratios do not significantly depend on the hard scale $Q^2$, and the outcomes are consistent with the various experimental data such as HERMES, BCDMS, JLab, SLAC, NMC, and EMC. Especially, in the mid-range of Bjorken $x$ values, the results are so desirable. Like our previous works for the $^4He$ and $^6Li$ nuclei, we again observe that at a fixed scale $Q^2$, the LO and the NLO EMC ratios with high precision are approximately the same. Therefore, one can conclude that at a given hard scale, the LO approximation is good enough for calculating the EMC ratios of light nuclei.

In this paper, using the stochastic modeling of the non-equilibrium statistical mechanics in the momentum space, the evolution equations of the parton distribution functions (PDF) usually used in the hadrons phenomenology are generated. These stochastic modeling PDF evolution equations are the same as those of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) ones, but they can be obtained by a more simplistic mathematical procedure based on the non-equilibrium statistical mechanics and the theory of Markov processes.

The charm structure function, ($F_2^{c\overline{c}}(x, Q^2)$), is calculated in the framework of the $k_t$-factorization formalism by using the unintegrated parton distribution functions (UPDF), which are generated through the Kimber et al. (KMR) and Martin et al. (MRW) procedures. The Martin group (MMHT2014) parton distribution functions (PDF) is used as the input PDF for the corresponding UPDF. The resulted $F_2^{c\overline{c}}(x, Q^2)$ is compared with the predicted data and the calculations given by the ZEUS and H1 collaborations, the parton pQCD theory, i.e. the general-mass variable-flavour-number scheme (GMVFNS), the LO collinear procedure and the saturation model introduced by Golec-Biernat and W\"usthoff (GBW), respectively. In general, it is shown that the calculated charm structure functions based on the stated above two UPDF schemes are consistent with the experimental data and other theoretical predictions. Also, a short discussion is presented regarding the KMR and MRW UPDF behaviors.

In this work, the P-P(\={P}) Drell-Yan lepton pair production (DY) differential cross sections at hadrons colliders, such as LHC and TEVATRON, are studied in the ${k_{t}}$-factorization framework. In order to take into account the transverse momenta of incoming partons, we use the unintegrated parton distribution functions (UPDF) of Kimber et al (KMR) and Martin et al (MRW) in the leading order (LO) and next-to-leading-order (NLO) levels with the input MMHT2014 PDF libraries. Based on the different off-shell partonic matrix elements, we analyze the behaviors of DY differential cross sections with respect to the invariant mass, the transverse momentum and the rapidity as well as the specific angular correlation between the produced leptons. The numerical results are compared with the experimental data, in different energies, which are reported by various collaborations, such as CDF, CMS, ATLAS and LHCb. It is shown that the NLO-MRW and KMR schemes predict closer results to the data compared to the LO-MRW, since we do not have fragmentation., Comment: To be published in Phys. Rev. D

In the present paper, we check and study the validity of the \textsc{KaTie} parton level event generator, by calculating the inclusive electron-proton (ep) dijet and the Proton-proton (p-p) Drell-Yan electron-pair productions differential cross sections in the $k_t$- and collinear factorization frameworks. The Martin-Ryskin-Watt (MRW) unintegrated parton distribution functions (UPDFs) are used as the input UPDFs. The results are compared with those of ZEUS ep inclusive dijet and ATLAS p-p Drell-Yan electron-pair productions, experimental data. The \textsc{KaTie} parton level event generator can directly calculate the cross sections in the $k_t$-factorization framework. It is noticed that the lab to the Breit transformation in this generator is not correctly implemented by its author, so the produced output does not cover the ep ZEUS experimental data in which the mentioned transformation is applied. By fixing the above transformation in the \textsc{KaTie} generator code, we could appropriately produce the ep inclusive dijet differential cross section, in comparison with those of ZEUS data. It is also shown that the MRW at the NLO level, with the angular ordering constraint can successfully predict the ATLAS p-p Drell-Yan data. Finally, as it is expected, we conclude that the $k_t$-factorization is an appropriate tool for the small longitudinal parton momenta and high center of mass energies, with respect to the collinear one.

In this paper, we investigate the differential cross sections of the inclusive jet and dijet productions of the ZEUS collaboration data at the center of mass energies of $\sim 300\;GeV$ and $ 319\;GeV$ using the $k_t$ and $(z,k_t)\textrm{-factorizations}$ with the different unintegrated and double unintegrated parton distribution functions, i.e., UPDFs and DUPDFs, respectively. The \textsc{KaTie} event generator is used to calculate the differential cross section with the UPDFs, while for the input DUPDFs the calculations are directly performed by evaluating the corresponding matrix elements. We check the effect of choosing the different implementation of angular or strong ordering constraints using the UPDFs and the corresponding DUPDFs of Kimber-Martin-Ryskin (KMR) and the leading-order (LO) and next-to-leading-order (NLO) Martin-Ryskin-Watt (MRW) approaches. The impacts of choosing virtualities $k^2 = k_t^2$ or $k^2 = {k_t^2\over {(1-z)}}$ in the differential cross section predictions for the ZEUS experimental data are also investigated. It is observed that, as one should expect, the applications of $(z,k_t)\textrm{-factorization}$ is better than the $k_t\textrm{-factorization}$ framework for the predictions of high virtuality $Q^2$ with respect to the ZEUS collaboration data, and also the results of the KMR and LO-MRW UPDFs and DUPDFs are reasonably close to each other and in general can describe the data. It is also observed that only in the case of $k_t$-factorization, the inclusion of Born level makes our results to overshoot the inclusive jet experimental data.

In the present work, it is intended to calculate and study the single and double differential cross-sections of the prompt single-photon production as a function of produced single-photon transverse momentum and rapidity, in the high-energy P-P(\={P}) colliders, such as LHC and TEVATRON. The differential cross-sections of prompt single-photon production are calculated in the $k_t$-factorization frameworks using various angular ordering unintegrated parton distribution functions (UPDF), namely the Kimber et al. and Martin et al. procedures. These scheme-dependent UPDF are generated in the leading and next-to-leading order levels to predict and analyze the different partonic contributions to the above cross-sections. The above two procedures utilize the phenomenological parton distribution functions (PDF) libraries of Martin et al., i.e., MMHT2014. It is shown that the calculated prompt single-photon production differential cross-sections in the above frameworks are relatively successful in generating satisfactory results compared to the experimental data of different collaborations, i.e., CDF (2017), ATLAS (2017), CMS (2011) and D0 (2006), as well as the other theoretical predictions such as collinear factorization Monte Carlo calculations (the JETPHOX, SHERPA, PYTHIA, and MCFM methods). Also, for a closer precision, the differential cross-section for the NLO gluon-gluon , quark-gluon and quark-(anti)quark processes are calculated. An extensive discussion and comparison are made regarding (i) the behavior of the contributing partonic sub-processes, (ii) the possible double-counting between the $2\to 2$ and $2\to 3$ sub-processes, i.e., gluons-fusion, in our calculated prompt single-photon production differential cross-sections and (iii) the sensibility check of our results to the different angular ordering constraints.

Recently, the ATLAS data of isolated three-photon production showed that the next-to-leading order (NLO) collinear factorization is not enough to describe experimental data. Therefore, one needs to calculate the cross section beyond the NLO, and as showed later, these data can be well described by the NNLO calculation within the collinear factorization framework. However, it is shown that the $k_t$-factorization can be quite successful in describing exclusive and high energy collision processes, henceforth we decided to calculate isolated three-photon production within this framework. In this work we use the Martin, Ryskin, and Watt unintegrated parton distribution functions (MRW UPDFs) at LO and NLO levels, in addition to parton branching (PB) UPDFs in order to calculate cross section which we utilize the KATIE parton level event generator. It will be shown that in contrast to collinear factorization, the $k_t$-factorization can describe quiet well the three-photon production ATLAS data. Interestingly our results using the NLO-MRW and PB UPDFs can cover the data within their uncertainty bands, similar to the NNLO collinear results.

We previously investigated the compatibility of the $Kimber$-$Martin$-$Ryskin$ ($KMR$) $unintegrated$ parton distribution functions ($UPDF$) with the experimental data on the proton (longitudinal) structure functions ($PSF$ ($PLSF$)). Recently $Golec-Biernat$ and $Stasto$ ($GBS$) demonstrated that the differential version of $KMR$ prescription and the implementations of angular (strong) ordering ($AOC$ ($SOC$)) constraints, cause the negative-discontinuous $UPDF$ with the ordinary parton distribution functions ($PDF$) as the input, which leads to a sizable effect on the calculation of $PSF$. In the present work, we use the new $MMHT2014$-$LO$-$PDF$ as the input and focus on the $UPDF$ behaviors as was raised by $GBS$. The resulting $PSF$ and $PLSF$ are compared with the $MSTW2008$-$LO$-$PDF$ and $MRST99$-$PDF$ and the 2014 data given by the $ZEUS$ and $H1$ collaborations. The calculated $PSF$ and $PLSF$ based on the integral prescription of the $KMR$-$UPDF$ with the $AOC$ and the ordinary $PDF$ as the input are reasonably consistent with the experimental data. Therefore, they are approximately independent to the $PDF$ (no need to impose cutoff on the $PDF$). At very small $x$ regions because of the excess of gluons in the $MMHT2014$-$LO$-$PDF$ and $MSTW2008$-$LO$-$PDF$, an increase in $PSF$ and $PLSF$ is achieved. Finally, according to the $GBS$ report the differential version by using the cutoff independent $PDF$ produces results far from the experimental data., Comment: 7 figures

The inductive charging of electric vehicles requires specific measurement and calibration systems. In fact, the measurement of power on board involves DC signals, which are superimposed to a significant AC ripple up to or over 150 kHz, depending on the type of charging system. A calibration method that makes use of a phantom power, based on two independent but synchronized circuits, is considered, simulating the charging voltage and current. This paper describes in detail a solution in the realization of the voltage calibration chain, based on the use of a DC voltage calibrator, an injector and a voltage divider., Comment: 2 pages, Conference on Precision Electromagnetic Measurements (CPEM 2018), Paris

The averaged effective two-body interaction (\textit{AEI}) which can be generated through the lowest order constrained variational (\textit{LOCV}) method for symmetric nuclear matter (\textit{SNM}) with the input \textit{Reid}68 nucleon-nucleon potential, is used as the effective nucleon-nucleon potential in the folding model to describe the heavy-ion (\textit{HI}) elastic scattering cross sections. The elastic scattering cross sections of $^{12}C-^{12}C$ and $^{16}O-^{16}O$ systems are calculated in the above frameworks. The results are compared with the corresponding calculations coming from the fitting procedures with the input finite range \textit{DDM3Y1-Reid} potential and the available experimental data at different incident energies. It is shown that a reasonable description of the elastic $^{12}C-^{12}C$ and $^{16}O-^{16}O$ scattering data at the low and the medium energies can be obtained by using the above \textit{ LOCV AEI}, without any need to define a parameterize density dependent function in the effective nucleon-nucleon potential, which is formally considered in the typical \textit{DDM3Y1-Reid} interactions.

The unintegrated parton distribution functions (UPDFs) of $^3H$, $^3He$ and $^4He$ nuclei are generated to calculate their structure functions (SFs) in the $k_t$-factorization approach. The Kimber-Martin-Ryskin (KMR) formalisn is applied to evaluate the double-scale UPDFs of these nuclei from their single-scale parton distribution functions (PDFs), which can be obtained from the constituent quark exchange model (CQEM). Afterwards, these SFs are used to calculate the European Muon Collaboration (EMC) ratios of these nuclei. The resulting EMC ratios are then compared with the available experimental data and good agreement with data is achieved. In comparison with our previous EMC ratios, in which the conventional PDFs were used in the calculations, the accord of the present outcomes with experiment at the small $x$ region becomes impressive. Therefore, it can be concluded that the $k_t$ dependence of partons can reproduce the general form of the shadowing effect at the small $x$ values in above nuclei.

In the present work, it is intended to calculate the unintegrated parton distribution functions (UPDFs) of the $^6{Li}$ nucleus, which depend not only on the longitudinal momentum fraction $x$ (the Bjorken variable) and the factorization scale $\mu$ of partons, but also on their transverse momentum ($k_t$). Therefore, the KMR and MRW procedures are applied to generate the $k_t$-dependent parton distributions from the familiar integrated parton densities (PDFs), which were determined in our recent related work, using the constituent quark exchange model (CQEM) for the $^6Li$ nucleus. Then, the resulting UPDFs of $^6{Li}$ are compared with the UPDFs of free proton from our previous work. Afterwards, from the $k_t$-factorization formalism, the structure function (SF) of $^6{Li}$ nucleus is computed to extract its European Muon Collaboration (EMC) ratio. The results are compered with those generated from the $^6Li$ nucleus PDFs and the available NMC experimental data. It is observed that, especially in the small $x$ region, where the $k_t$ dependence of partons are important in the calculations of the hadron structure functions, the present EMC ratios are extremely improved compared with those of our previous work, and show an excellent agreement with the NMC experimental data.

In the present work, which is based on the k_-factorization framework, it is intended to make a detail study of the isolated prompt-photon pairs (IPPP) production in the high-energy inelastic hadron-hadron collisions differential cross section. The two scheme-dependent unintegrated parton distribution functions (UPDF) in which the angular ordering constraints (AOC) are imposed, namely the Kimber-Martin-Ryskin (KMR) and the Martin-Ryskin-Watt (MRW) approaches, in the leading and the next-to-leading orders (LO and NLO) are considered, respectively. These two prescriptions (KMR and MRW) utilize the phenomenological parton distribution functions (PDF) libraries of Martin et al, i.e. the MMHT2014. The computations are performed in accordance with the initial dynamics of latest existing experimental reports of the D0, CDF, CMS and ATLAS collaborations and the different experimental constraints. It is shown that above frameworks are capable of producing acceptable results, compared to the experimental data, the pQCD and some Monte Carlo calculations (i.e. 2\gamma-NNLO, SHERPA, DIPHOX and RESBOS). It is also concluded that the KMR framework produces better results in the higher center-of-mass energies, while the same thing can be argued about the LO-MRW prescription in lower energies. Additionally, these two schemes show different behavior in the regions where the fragmentation and higher pQCD effects become important. A clear prediction for the various shoulders and tails which were detected experimentally are observed and discussed in the present theoretical approaches. The possible double countings between 2 \rightarrow 2 and 2 \rightarrow 3 processes are studied., Comment: 16 Figures

The present work is devoted to study the high-energy $QCD$ events, such as the di-jet productions from proton-proton inelastic collisions at the $LHC$ in the forward-center and the forward-forward configurations, using the $unintegrated$ parton distribution functions ($UPDF$) in the $k_t$-factorization framework. The $UPDF$ of $Kimber$ et. al. ($KMR$) and $Martin$ et.al. ($MRW$) are generated in the leading order ($LO$) and next-to-leading order ($NLO$), using the $Harland-Lang$ et al. ($MMHT2014$) $PDF$ libraries. While working in the forward-center and the forward-forward rapidity sectors, one can probe the parton densities at very low longitudinal momentum fractions ($x$). Therefore, such a computation can provide a valuable test-field for these $UPDF$. We find very good agreement with the corresponding di-jet production data available from $LHC$ experiments. On the other hand, as we have also stated in our previous works, (i.e. the protons longitudinal and transverse structure function as well as hadron-hadron $LHC$ $W/Z$ production), the present calculations based on the $KMR$ prescriptions show a better agreement with the corresponding experimental data. This conclusion is achieved, due to the particular visualization of the angular ordering constraint ($AOC$), despite the fact that the $LO-MRW$ and the $NLO-MRW$ formalisms both employ better theoretical descriptions of the $Dokshitzer$-$Gribov$-$Lipatov$ -$Altarelli$-$Parisi$ ($DGLAP$) evolution equation, and hence are expected to produce better results. The form of the $AOC$ in the $KMR$ prescription automatically includes the re-summation of the higher-order $ln({1/x})$ type contributions, i.e. the $Balitski$-$Fadin$-$Kuraev$-$Lipatov$ ($BFKL$) logarithms, in the $LO$-$DGLAP$ evolution equation.

Quit recently, two sets of new experimental data from the $LHCb$ and the $CMS$ collaborations have been published, concerning the production of the $Z^0$ vector boson in hadron-hadron collisions with the center-of-mass energy $E_{CM}= \sqrt{s}=13\;TeV$. On the other hand, in our recent work, we have conducted a set of $NLO$ calculations for the production of the electroweak gauge vector bosons, utilizing the unintegrated parton distribution functions ($UPDF$) in the frameworks of $Kimber$-$Martin$-$Ryskin$ ($KMR$) or $Martin$-$Ryskin$-$Watt$ ($MRW$) and the $k_t$-factorization formalism, concluding that the results of the $KMR$ scheme are arguably better in describing the existing experimental data, coming from $D0$, $CDF$, $CMS$ and $ATLAS$ collaborations. In the present work, we intend to follow the same $NLO$ formalism and calculate the rate of the production of the $Z^0$ vector boson, utilizing the $UPDF$ of $KMR$ within the dynamics of the recent data. It will be shown that our results are in good agreement with the new measurements of the $LHCb$ and the $CMS$ collaborations., Comment: 3 figures

In a series of papers, we have investigated the compatibility of the $Kimber$-$Martin$-$Ryskin$ ($KMR$) and $Martin$-$Ryskin$-$Watt$ ($MRW$) $unintegrated$ parton distribution functions ($UPDF$) as well as the description of the experimental data on the proton structure functions. The present work is a sequel to that survey, via calculation of the transverse momentum distribution of the electro-weak gauge vector bosons in the $k_t$-factorization scheme, by the means of the $KMR$, the $LO\;MRW$ and the $NLO\;MRW$ $UPDF$, in the next-to leading order ($NLO$). To this end, we have calculated and aggregated the invariant amplitudes of the corresponding $involved$ diagrams in the $NLO$, and counted the individual contributions in different frameworks. The preparation process for the $UPDF$ utilizes the $PDF$ of $Martin$ et al, $MSTW2008-LO$, $MSTW2008-NLO$, $MMHT2014-LO$ and $MMHT2014-NLO$ as the inputs. Afterwards, the results have been analyzed against each other, as well as the existing experimental data. Our calculation show excellent agreement with the experiment data. It is however interesting to point-out that, the calculation using the $KMR$ framework illustrates a stronger agreement with the experimental data, despite the fact that the $LO\;MRW$ and the $NLO\;MRW$ formalisms employ a better theoretical description of the $DGLAP$ evolution equation. This is of course due to the use of the different implementation of the angular ordering constraint in the $KMR$ approach, in which automatically includes the re-summation of $ln({1/x})$, $BFKL$ logarithms, in the $LO$-$DGLAP$ evolution equation.

To overcome the complexity of generalized two hard scale ($k_t$,$\mu$) evolution equation, well known as the $Ciafaloni$, $Catani$, $Fiorani$ and $Marchsini$ ($CCFM$) evolution equations, and calculate the unintegrated parton distribution functions ($UPDF$), $Kimber$, $Martin$ and $Ryskin$ ($KMR$) proposed a procedure based on ($i$) the inclusion of single-scale ($\mu$) only at the last step of evolution and ($ii$) the angular ordering constraint ($AOC$) on the $DGLAP$ terms (the $DGLAP$ collinear approximation), to bring the second scale, $k_t$ into the $UPDF$ evolution equations. In this work we intend to use the $MSTW 2008$ (Martin et al) parton distribution functions (PDF) and try to calculate $UPDF$ for various values of $x$ (the longitudinal fraction of parton momentum), $\mu$ (the probe scale) and $k_t$ (the parton transverse momentum) to see the general behavior of three dimensional $UPDF$ at the $NLO$ level up to the $LHC$ working energy scales ($\mu^2)$. It is shown that there exits some pronounced peaks for the three dimensional $UPDF$ $(f_a(x,k_t))$ with respect to the two variables $x$ and $k_t$ at various energies ($\mu$). These peaks get larger and move to larger values of $k_t$, as the energy ($\mu$) is increased. We hope these peaks could be detected in the $LHC$ experiments at $CERN$ and other laboratories in the less exclusive processes.

The charm quark structure function $F^c_2$ and the longitudinal structure function $F_l^p$ are directly sensitive to the gluon content of proton and therefore are crucial in understanding of proton structure function, in particular at low momentum transfer $Q^2$ and low Bjorken x. In the framework of perturbative QCD the charm structure function is calculated in the leading order (LO) and the proton structure function is investigated in the next leading order (NLO) at small x region. The valence quark distribution is obtained from the relativistic quark-exchange model. The calculated $F_2^c(x,Q^2)$, $F_2^p(x,Q^2)$ and $F_L^p(x,Q^2)$, are compared with the present available experimental data., Comment: 4 Pages, 4 Figures, Presented at IPM-LHP May 2006

By using the quark-exchange formalism, realistic Faddeev wave functions and Fermi motion effect, we investigate deep inelastic electron scattering from A=3 mirror nuclei in the deep-valence region. The initial valence quarks in put are taken from the GRV's next-to-leading order calculations on $F_2^p(x,Q^2)$ which give very good fit to the available data in the $(x,Q^2)$-plane. It is shown that the free neutron to proton structure functions ratio can be extracted from corresponding EMC ratios for $^3He$ and $^3H$ mirror nuclei by using self-consistent iteration procedure and the results are in good agreement with other theoretical models as well as present available experimental data, especially the one expected form the proposed 11 GeV Jefferson Laboratory., Comment: 16 pages, 5 figures

The method of lowest-order constrained variational, which predicts reasonably the nuclear matter semi-empirical data is used to calculate the equation of state of beta-stable matter at finite temperature. The Reid soft-core with and without the N-$\Delta$ interactions which fits the N-N scattering data as well as the $UV_{14}$ potential plus the three-nucleon interaction are considered in the nuclear many-body Hamiltonian. The electron and muon are treated relativistically in the total Hamiltonian at given temperature, to make the fluid electrically neutral and stable against beta decay. The calculation is performed for a wide range of baryon density and temperature which are of interest in the astrophysics. The free energy, entropy, proton abundance, etc. of nuclear beta-stable matter are calculated. It is shown that by increasing the temperature, the maximum proton abundance is pushed to the lower density while the maximum itself increases as we increase the temperature. The proton fraction is not enough to see any gas-liquid phase transition. Finally we get an overall agreement with other many-body techniques, which are available only at zero temperature., Comment: LaTex, 20 pages

It is shown that the method of lowest order contrained variational (LOCV) which is based on the cluster expansion theory is a reliable many-body technique to calculate the nuclear matter equation of states. In this respect the state-dependence correlation functions and effective interactions which have been produce by LOCV calculation with the Reid Soft Core and $\Delta $-Reid interactions are used to estimate the size of higher order cluster terms such as the effect of three body cluster energy in nuclear matter. We find that the three-body cluster energy is less than 1 MeV beyond the nuclear matter saturation density and it has weaker density dependence than our previous calculation with the state-averaged coorrelation functions and effective interactions. Finally, we conclude that the LOCV method is good enough to calculate other propertiesn of quantal fluids., Comment: 15 pages, 8 figures