Two-pion Bose-Einstein correlations in pp collisions at $\sqrt{s}=900$ GeV

We report on the measurement of two-pion correlation functions from pp collisions at $\sqrt{s}=900$ GeV performed by the ALICE experiment at the Large Hadron Collider. Our analysis shows an increase of the HBT radius with increasing event multiplicity, in line with other measurements done in particle- and nuclear collisions. Conversely, the strong decrease of the radius with increasing transverse momentum, as observed at RHIC and at Tevatron, is not manifest in our data.


I. INTRODUCTION
Proton-proton collisions at √ s = 900 GeV have been recorded by ALICE (A Large Ion Collider Experiment) at the Large Hadron Collider (LHC) at CERN [1].Hadron collisions at these energies provide an opportunity to probe Quantum Chromodynamics (QCD) under extreme conditions.The distinguishing feature of QCD is the mechanism of color confinement, the physics of which is not fully understood, due in part to its theoretical intractability [2].The confinement mechanism has a physical scale on the order of the proton radius and is especially important at low momentum.
Bose-Einstein enhancement of identical-pion pairs at low relative momentum was first observed in p p collisions by Goldhaber, Goldhaber, Lee and Pais 50 years ago [3].Since then, two-pion correlations have been successfully applied to assess the spatial scale of the emitting source in e + e − [4], hadron-hadron and lepton-hadron [5], and heavy ion [6] collisions.Especially in the latter case, this technique, known as Hanbury Brown -Twiss (HBT) interferometry [7,8] and being a special case of femtoscopy [9,10], has been developed into a precision tool to probe the dynamically-generated geometry of the emitting system.In particular, a first order phase transition between the color-deconfined and -confined states was precluded by the observation of short timescales [6].At the same time, femtoscopic measurement of bulk collective flow, manifesting itself via dynamical dependences of femtoscopic scales ("homogeneity lengths" [11,12]), provided hints that a strongly self-interacting system was created in the collision.This was further corroborated by the positive correlation between the HBT radius and the multiplicity of the event [6].
In particle physics, overviews of femtoscopic measurements in hadron-and lepton-induced collisions [4,5,13] reveal systematics surprisingly similar to those mentioned above for heavy-ion collisions.Moreover, in the first direct comparison of femtoscopy in heavy-ion collisions at RHIC, and proton collisions in the same apparatus, a virtually identical multiplicity-and momentum-dependence was reported in the two systems [14].
A systematic program of femtoscopic measurements in pp and heavy-ion collisions at the LHC will shed considerable light on the nature, the similarities, and the differences of their dynamics.With the present work, we begin this program.

II. EXPERIMENT AND DATA ANALYSIS
The data discussed in this article were collected in December 2009, during the first stable-beam period of the LHC commissioning.The two beams were at the LHC injection energy of 450 GeV and each had 2-4 bunches, one of them colliding at the ALICE intersection point.The bunch intensity was typically 5 × 10 9 protons, giving a luminosity of the order of 10 26 cm −2 s −1 and a rate for inelastic proton-proton collisions of a few Hz.
Approximately 3 × 10 5 minimum bias pp collision events were identified by signals measured in the forward scintilla-tors (VZERO) and the two layers of the Silicon Pixel Detector (SPD) [15].The VZERO counters are placed on either side of the interaction region at z = 3.3 m and z = -0.9 m.They cover the region 2.8 < η < 5.1 and −3.7 < η < −1.7 and record both amplitude and time of signals produced by charged particles.The minimum-bias trigger required a hit in one of the VZERO counters or in one of the two SPD layers which cover the central pseudorapidity regions |η| < 2 (inner) and |η| < 1.4 (outer).The events were collected in coincidence with the signals from two beam pick-up counters, one on each side of the interaction region, indicating the presence of passing bunches.The trigger selection efficiency for inelastic collisions was estimated to be 95-97% [16].
The VZERO counters were used also to discriminate against beam-gas and beam-halo events by requiring a strict matching between their timing signals (see Ref. [1] for details).This background was also rejected by exploiting the correlation between the number of clusters of pixels and the number of tracklets pointing to a reconstructed vertex.After these selections the fraction of background events remaining in the sample of events with at least one charged particle track was estimated to be below 0.1%.The trigger and run conditions are discussed in detail in Ref. [16].
The 250 k events used in the analysis were required to have a primary vertex (collision position) within 10 cm of the center of the 5 m long Time Projection Chamber (TPC) [17].This provides almost uniform acceptance for particles within the pseudorapidity range |η| < 0.8 for all events in the sample.Within this sample, we have selected events based on the measured charged-particle multiplicity M. The three multiplicity classes were M ≤ 6, 7 ≤ M ≤ 11, and M ≥ 12; about 70% of all events were falling into the first multiplicity class.The tracks used in determining the multiplicity were the same as those used for correlation analysis (see below) except that particle identification cuts were not applied.The measured multiplicity was converted to the charged-particle pseudorapidity density dN ch /dη by normalizing it to the pseudorapidity acceptance and by correcting it for the reconstruction efficiency and contamination.The correction factor was determined from a Monte Carlo simulation with the PHOJET event generator [18,19] and with the full description of the AL-ICE apparatus and is 0.71, 0.78, and 0.81, respectively, for the three multiplicity bins.The estimated systematic error is below 4%.The average charged-particle pseudorapidity density of the analyzed event sample is dN ch /dη =3.6.An alternative method based on SPD tracklets [16] gave the same result.
The ALICE Time Projection Chamber (TPC) [17] was used to record charged particle tracks as they leave ionization trails in the Ne-CO 2 -N 2 gas.The ionization electrons drift up to 2.5 m to be measured on 159 pad rows; the position resolution is better than 2 mm.Combined with a solenoidal magnetic field of B=0.5 T this leads to a momentum resolution ∼ 1% for pions with p T < 1 GeV/c.The ALICE Inner Tracking System (ITS) has also been used for tracking.It consists of six silicon layers, two innermost pixel detectors, two layers of drift detectors, and two outer layers of strip detectors, which provide up to six space points for each track.The tracks used in this analysis were reconstructed using the information from both the TPC (signals from at least 90 pad rows required) and the ITS.Separate studies have been done with TPC-only and ITS-only tracks, and were found to give results consistent with the combined ITS+TPC analysis.The tracks were required to project back to the primary interaction vertex within 0.2 cm (2.4 cm) in the transverse plane and 0.25 cm (3.2 cm) in longitudinal direction, if ITS+TPC (TPC only) information is used, thereby rejecting most secondary pions from weak decays.The pion tracks used in the correlation analysis had transverse momenta between 0.15 GeV/c and 1.0 GeV/c.ALICE provides excellent particle identification capability.In this analysis the particle identification was achieved by correlating the magnetic rigidity of a track with its specific ionization (dE/dx) in the TPC gas.The dE/dx of the TPC was calibrated using cosmic rays and its resolution was shown to be better than 5.5%, the design value.The contamination of the pion sample is negligible within the momentum range of 0.25 GeV/c < p < 0.65 GeV/c.Below and above this range it is on the order of 5% and is caused by electrons and kaons, respectively.

III. TWO-PION CORRELATION FUNCTIONS
The two-particle correlation function is defined as the ratio C (q) = A (q) /B (q), where A (q) is the measured distribution of pair momentum difference q = p 2 − p 1 , and B (q) is a similar distribution formed by using pairs of particles from different events (event mixing) [20].The limited statistics available (520 k identical-pion pairs with q inv < 0.5 GeV/c) allowed us to perform a detailed analysis only for the one-dimensional two-pion correlation function C(q inv ).The q inv is, for identical mass particles, equal to the modulus of the momentum difference |q| in the pair rest frame.
The correlation functions were studied in bins of event multiplicity and of transverse momentum, defined as half of the vector sum of the two transverse momenta, k T = |p T,1 +p T,2 |/2.During event mixing, all pion tracks from one event were paired with all pion tracks from another event.Every event was mixed with five other events with similar multiplicities; ten multiplicity bins were introduced for this purpose.The multiplicity binning improved the flatness of the correlation function at q inv > 1.5 GeV/c.Binning events according to their vertex position, on the other hand, had no effect on the correlation function and therefore was not used.Alternatively to event mixing, the denominator can be obtained by rotating one of the two tracks by 180 o in azimuth.The correlation functions obtained using this technique are generally flatter at high q inv than those from event mixing.The difference between the results obtained utilizing the two techniques was used in estimating the systematic errors.
For the correlation structures measured here, with characteristic widths ∼ 0.2 GeV/c, track splitting and track merging in the event reconstruction are small effects overall.Their impact on the results was carefully studied with the Monte Carlo simulation and turned out to be negligible.
Another apparatus effect considered is the momentum resolution.Momentum smearing for single particles has similar effect on the correlation structures in two-particle correlations i.e. it smears the correlation peak, making it appear lower and wider.We have studied this effect with the Monte-Carlo simulation of the ALICE detector and have found that for the width of the correlation peak expected here the effect is on the order of 1%.
Fig. 1 presents two-pion correlation functions measured by ALICE in pp collisions at √ s = 900 GeV, as a function of event multiplicity and transverse momentum k T .The denominator of the correlation function was obtained via event mixing and normalized such that the numbers of true and mixed pairs with 0.4 GeV/c < q inv < 0.6 GeV/c were equal.The q inv range used for normalization was chosen to be outside of the Bose-Einstein peak but as close as possible to it.The normalized distributions of positive and negative pion pairs were added together before building the ratio of true and mixed pairs.The Bose-Einstein enhancement is manifest at low q inv .A slight decrease of the correlation peak width is seen as multiplicity grows.The k T dependence is less obvious because 0 0.2 0.4 0.6 0.  the correlation baseline -the underlying two particle correlation without any Bose-Einstein enhancement -is systematically changing its shape between the low and high transverse momenta.
The correlation functions were fitted by a function accounting for the Bose-Einstein enhancement and for the mutual Coulomb interaction between the two particles: with λ describing the correlation strength and R inv being the Gaussian HBT radius [21].The factor K is the Coulomb function integrated over a spherical source of the size 1 fm.It is attenuated by the same factor λ as the Bose-Einstein peak.The factor D(q inv ) accounts for long-range correlations, like those arising from jets and/or from energy and momentum conservation, and plays an important role in the analysis as will be discussed later.
Neglecting the Coulomb interaction K(q inv ) ≡ 1 the fit function reduces to The difference between the R inv values obtained with and without the Coulomb correction is less than 0.05 fm.
While the Gaussian fit captures the bulk scales of the correlation, at low q inv the data points lie above the fit line.This feature was observed previously in pion correlations from particle collisions.An exponential fit matches the data better.However, contrary to the Gaussian R inv , the R inv parameter from Eq. ( 3) does not have a straightforward interpretation as the "radius of the source".We have used both functional forms and leave a detailed investigation of the correlation peak shape to future studies.In order to make the connection to established systematics at lower energy particle and heavy-ion collisions, a careful treatment of the long-range correlations, visible as a slope in the baseline of the correlation developing with increasing transverse momentum and represented by the factor D(q inv ) in Eqs.(1-3), is crucial.
In order to better understand the shape of the correlation baseline we have calculated correlation functions for pp collisions events generated by the model PHOJET and propagated through the ALICE detectors, performing an identical analysis for the simulated events as for the measured ones.The results are shown as open circles in Fig. 1.The model does not contain the Bose-Einstein effect, hence the lack of the peak at low q inv is expected.At low k T and low multiplicity, the model predicts a flat correlation function.However, as k T increases, long-range correlations start becoming visible as a distortion of the correlation function baseline similar to that seen in the experimental data.
The accuracy of our simulation in describing the correlation baseline was verified with unlike-sign pion pairs.The multiplicity and k T dependence of the π + π − functions is shown in Fig. 2. Correlation structures for non-identical pions include a mutual Coulomb interaction peak, here limited to the first bin at lowest q inv , and peaks coming from meson decays which should be correctly modeled in the event generator.Therefore, one can directly compare simulations with data.In Fig. 2, the simulated correlation functions agree reasonably well with the experimental data.This suggests that the same model (PHO-JET) can be used as a reasonable estimate also for identical particles to describe the correlation baseline under the Bose-Einstein peak.The presence of resonance peaks (like the K 0 S one at q inv = 412 MeV/c) and the fact that the simulated correlations for identical and non-identical pion pairs have different slopes, on the other hand, indicate that unlike-sign pion pairs cannot be directly used for the denominator of the identical pion correlations.
The procedure employed to extract the HBT radii with Eq. (1) using the PHOJET baseline is as follows.First, the simulation points shown in Fig. 1 are fitted with the 2nd-order polynomial Subsequently, the experimental correlation function is fitted by Eq. ( 1), taking the D(q inv ) from the PHOJET fit and adjusting λ and R inv .The two fits are represented in Fig. 1 by the lines going through the simulation and experiment data points, respectively.
In order to estimate the systematic error from the baseline determination we repeated the fitting procedure using a simulation performed with the PYTHIA [22] generator (version 6.4.21,Perugia-0 (320) tune [23]) instead of PHOJET.The HBT radii obtained in the two ways differ by up to 10%.In the following we use the average between them and we estimate the systematic error related to the baseline shape assumption to be half of the difference.
It is interesting to see what happens with the radii if the slope of the baseline is neglected.Assuming a flat baseline D(q inv ) ≡ a and treating a as the third fit parameter in Eq. ( 1) leads to R inv values that are similar to those obtained with the PHOJET or PYTHIA baseline at low k T values but smaller by up to 30% at high transverse momenta.This is because the broad enhancement caused by long-range correlations will be attributed to Bose-Einstein correlations, giving rise to smaller radii (wider correlation function).The resulting apparent k T dependence will be discussed in Section V.
The R inv obtained from the fit (the two highest multiplicity bins combined) is shown in Fig. 3.In order to reduce the statistical errors and to compare to other experiments, in the following sections of this article we analyze separately the multiplicity and the transverse momentum dependences.

IV. MULTIPLICITY DEPENDENCE OF THE HBT RADIUS
The multiplicity dependence of the obtained HBT radius is shown in Fig. 4 and Table I.The analysis here was re- For comparison, open symbols, red stars, and green filled boxes represent the data taken at the ISR [24], RHIC [14], and Tevatron [25], respectively.

TABLE I. One-dimensional HBT radius in pp collisions at
√ s = 900 GeV determined using pion pairs with k T = 0.1-0.55GeV/c, k T = 0.32 GeV/c, as a function of the charged-particle pseudorapidity density at midrapidity.The radii were obtained using the Gaussian fit function defined by Eq. ( 1 stricted to the first three transverse momentum bins k T = 0.1-0.55GeV/c.The mean transverse momentum for pairs with q inv < 0.2 GeV/c is k T = 0.32 GeV/c.The HBT radii were obtained by using PHOJET and PYTHIA to estimate the shape of the baseline, as explained in the previous section.The systematic errors related to the baseline assumption reflect the difference between the two.The systematic error related to the choice of the normalization and/or fit range was estimated to be 5%.An additional downward systematic error of 13-20% accounts for the difference between the event mixing and the rotation denominator techniques.The shaded area represents the three systematic errors added in quadrature.
The charged-particle pseudorapidity density dN ch /dη of the lowest multiplicity bin was calculated excluding events with multiplicities M < 2 because these events do not contribute to the numerator of the correlation function.Including all events and including only events with at least one like-sign pair would shift the point by 0.8 to the left and to the right, respectively.
An increase of the HBT radius with multiplicity is observed, consistent with the hadron-hadron collision systematics above √ s ∼ 50 GeV [13].While the average transverse momentum is similar in all four data sets, other aspects of the analysis, e.g. the average orientation of the momentum difference vector, can differ so the trends, not the absolute values, should be compared.In heavy-ion collisions, this multiplicity dependence has been associated with the particle composition and overall volume of the final state system [6,26,27] or with final-state hadronic rescattering [28].The relation observed in heavy-ion collision data [6], R ∼ a + b(dN ch /dη) 1/3 , where a and b are constants, appears to be consistent with our data within our systematic errors.For high energy pp collisions, it has been suggested that a similar behavior could originate from final-state hadronic rescattering for short hadronization times [29].In an alternative scenario, the increase of the HBT radius with multiplicity results from the fact that the high multiplicity pp events mostly come from hard parton scattering, and the hadronization length, i.e. the distance travelled by a parton before hadronization, is roughly proportional to the parton energy [30].
The fitted correlation strength λ is lower than unity, the value expected for the ideal Bose-Einstein case.One reason for this is the non-Gaussian shape of the peak, caused at least partially by pions from decays of short-(∆, ρ) and longlived resonances (ω, η, η ′ ).On the detector side, λ can be reduced by the particle misidentification; this effect is however small in our data sample.In ALICE, λ decreases from 0.37±0.03 to 0.32±0.03between the lowest and the highest multiplicity, in close agreement with the E735 measurements at Tevatron [25].A similar trend was observed by UA1 in p p collisions at √ s = 630 GeV/c [31]; the fact that their λ values were lower may have to do with the lack of the particle identification and the resulting dilution of the correlation peak.In a final-state hadronic rescattering model [29], a correlation strength dropping with multiplicity in high-energy pp collisions was attributed to the increased contribution from long-lived resonances in higher multiplicity events.
An increase of the HBT radius with increasing particle multiplicity was recently reported by the CMS Collaboration for the same collision system and energy [32].The authors fit the correlation peak by an exponential (Eq.( 3)).An analogous approach in our case (Table II) yields radii that are rather similar to the Gaussian ones (Table I) once scaled down by √ π [32].In order to compare between the two experiments we perform a fit to an inclusive correla-tion (all multiplicities and k T 's).The exponential fit to the correlation functions obtained using event mixing and using rotation yields R inv =1.61±0.07(stat.)±0.05(syst.)fm and R inv =1.31±0.05(stat.)±0.22 (syst.)fm, respectively.This is in close agreement with the corresponding values quoted by CMS, 1.72±0.06fm and 1.29±0.04fm.

V. TRANSVERSE MOMENTUM DEPENDENCE OF THE HBT RADIUS
One of the key features of the bulk system created in nuclear collisions is its large collective flow.The fingerprint of this flow is a specific space-momentum correlation signature, revealed in the transverse momentum dependence of the Gaussian HBT radius [6].While quantitative comparison between particle and heavy ion studies is complicated by experiments using different acceptances and techniques, a recent comparison of the HBT radii from pp and Au+Au collisions at RHIC indicates an almost identical p T dependence between these collision systems [14].Again, this raises the interesting question whether hadron collisions at the highest energies already develop a bulk, collective behavior.
The k T dependence of our measured HBT radius is shown in Fig. 5.The choice of the fitting method, which only weakly affects the multiplicity dependence of the HBT radius discussed in the previous section, is of crucial importance for the transverse momentum dependence.Taking the baseline shape from the Monte Carlo leads to an HBT radius that is nearly independent of k T (filled black circles and red boxes for PHO-JET and PYTHIA, respectively).Assuming a flat baseline, on the other hand, results in a radius falling with k T (green stars).As discussed in the previous section, the experimental unlikesign pion correlation functions are close to the predictions of PHOJET and PYTHIA and we consider using the average between the two cases as baseline to be a reliable estimate for the HBT radii.
The radii obtained in this fashion are summarized in Tables III and IV and shown in Fig. 6 where we compare them to RHIC and Tevatron data [13].Like for the multiplicity dependence, the systematic error band represents a quadratic sum of the error related to the baseline assumption (0-10%), the fit range (10%), and the denominator construction method (mixing/rotating, 7-17%).The lowest-k T point is significantly below the RHIC and Tevatron results.It should be noted that the ALICE analysis was performed on a minimum bias event sample and the averaged charged-particle pseudorapidity density is dN ch /dη =3.6 while the Tevatron events are biased to high multiplicity, dN ch /dη =14.4,similar to our highest multiplicity bin.As visible in Fig. 3, the lowest-k T point at the high multiplicity is at R inv ≈1.2 fm, approaching the Tevatron points.The STAR results, on the other hand, were obtained from events with dN ch /dη = 4.3 i.e. similar to the ALICE case and thus a similar reasoning cannot explain the difference.
Two tests were performed to make sure that the low HBT radius value at low transverse momenta is not caused by apparatus effects.First, the analysis was repeated using only the ITS and thus reducing the low-momentum cut-off by about   6. One-dimensional Gaussian HBT radius in pp collisions at √ s = 900 GeV as a function of transverse momentum k T (full dots).The mean charged-particle multiplicity density was dN ch /dη =3.6.PHOJET simulation was used to determine the baseline of the correlations.Using PYTHIA and a flat baseline leads to systematic deviations up and down, respectively; the related systematic errors as indicated by the shaded area.Stars and filled boxes represent the radii measured at RHIC [14] and Tevatron [25], respectively.50 MeV.This analysis yielded the same HBT radius which demonstrates that the energy loss is not an issue.Second, as seen in Fig. 3 the low-k T point is mostly driven down by the contribution of the low multiplicity events.Since the vertex resolution in these events is worse this might in principle deteriorate the momentum resolution and smear out the correlation function peak.In order to test this the analysis was performed without using the event vertex constraint for momentum determination.The results, again, were unchanged.This, and the distinct K 0 S peak in the unlike-sign pion corre-lation functions in the low-multiplicity low-k T bin of Fig. 2, indicate that the momentum resolution is not spoiled in low multiplicity events.Even more important than the position of the first point, albeit related to it, is the question of the slope of the points in Fig. 6.Our measured HBT radius is practically independent of k T within the studied transverse momentum range.The slope crucially depends on the baseline shape assumption, as was shown in Fig. 5.The results from the experiments to which we are comparing in Fig. 6 were extracted using a flat background (although the STAR experiment also studied the effects of using other types of backgrounds for their data to account for the non-femtoscopic effects [14]).Assuming that PHOJET and PYTHIA are correct such a procedure may lead to a misinterpretation of the low-q enhancement of the correlation function, that is coming from long-range correlations (most probably mini-jet like), as a Bose-Einstein enhancement.As the impact of this may depend on the details of each experiment (certainly on the collision energy) we do not attempt to resolve this question quantitatively.However, we stress again the usefulness of non-identical pion correlation in constraining the correlation baseline.

VI. SUMMARY
In summary, ALICE has measured two-pion correlation functions in pp collisions at √ s = 900 GeV at the LHC.Consistent with previous measurements of high-energy hadronhadron and nuclear collisions, the extracted HBT radius R inv increases with event multiplicity.Less consistent is the relation between R inv and the pion transverse momentum where the ALICE measured HBT radius in minimum bias events is practically constant within our errors and within the transverse momentum range studied.
• Swedish Reseach Council (VR) and Knut & Alice Wallenberg Foundation (KAW); • Ukraine Ministry of Education and Science; • United Kingdom Science and Technology Facilities Council (STFC); • The United States Department of Energy, the United States National Science Foundation, the State of Texas, and the State of Ohio.

FIG. 1 .
FIG. 1. Correlation functions for identical pions from pp collisions at√ s = 900 GeV (full dots) and those obtained from a simulation using PHOJET (open circles).Positive and negative pion pairs were combined.The three columns represent collisions with different charged-particle multiplicities M; the transverse momentum of pion pairs k T (GeV/c) increases from top to bottom.The lines going through the points represent the Gaussian fits discussed in the text.

FIG. 2 .
FIG.2.One-dimensional correlation functions for π + π − pairs from pp collisions at √ s = 900 GeV.The columns and rows are defined as in Fig.1.

FIG. 3 .
FIG.3.Extracted HBT radius as a function of k T for low (black circles) and high (red squares) multiplicity events.The error bars are statistical.The shaded bands represent the systematic errors related to the baseline shape assumption and to the fit range, added quadratically.

FIG. 4 .
FIG. 4. One-dimensional Gaussian HBT radius in pp collisions at√ s = 900 GeV determined using pion pairs with k T = 0.1-0.55GeV/c, k T = 0.32 GeV/c, and shown as a function of the charged-particle multiplicity at midrapidity (full dots).The shaded band represents the systematic errors (see text).For comparison, open symbols, red stars, and green filled boxes represent the data taken at the ISR[24], RHIC[14], and Tevatron[25], respectively.

FIG. 5 .
FIG. 5. One-dimensional Gaussian HBT radius in pp collisions at √ s = 900 GeV as a function of transverse momentum k T .Three fitting methods, differing by the choice of the baseline parametrization, are compared. ).

TABLE II .
One-dimensional HBT radius in pp collisions at √ s = 900 GeV determined using pion pairs with k T = 0.1-0.55GeV/c,kT = 0.32 GeV/c, as a function of the charged-particle pseudorapidity density at midrapidity.The radii were obtained using the exponential fit function defined by Eq. (3).

TABLE III .
One-dimensional HBT radius in pp collisions at √ s = 900 GeV as a function of the pair k T .The radii were obtained using the Gaussian fit function defined by Eq. (1).

TABLE IV .
One-dimensional HBT radius in pp collisions at √ s = 900 GeV as a function of the pair k T .The radii were obtained using the exponential fit function defined by Eq. (3).