Bled Workshops in Physics Vol. 12, No. 1 p. 73 Scattering phase shift and resonance properties* S. Prelovseka,b, C. B. Langc and D. Mohlerd a J. Stefan Institute, Jamova 39,1000 Ljubljana, Slovenia b Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia c Institut für Physik, FB Theoretische Physik, Universitat Graz, A-8010 Graz Austria d TRIUMF, 4004 Wesbrook Mall Vancouver, BC V6T 2A3, Canada Abstract. We describe the method for extracting the elastic scattering phase shift from a lattice simulation at an introductory level, for non-lattice practitioners. We consider the scattering in a resonant channel, where the resulting phase shift 6(s) allows the lattice determination of the mass and the width of the resonance from a Breit-Wigner type fit. We present the method for the example of P-wave nn scattering in the p meson channel. 1 Introduction The determination of the strong decay width of a hadronic resonance in lattice QCD is a much more demanding task than the determination of its approximate mass. The only available method (that was applied up to now) was proposed by Luscher [1] and is rather indirect. It applies for the case when the resonance appears in the elastic scattering of two hadrons Hi H2 —> R —> Hi H2. • First, the energy spectrum En of the system of two interacting hadrons H1 H2 enclosed in a few-fermi box has to be determined. The system is illustrated in Fig. 1. The spectrum in a finite box En is discrete and few (one or two) lowest energy levels have to be determined by lattice simulation. • The shift of the energy En with respect to the non-interacting energy Ehi (pi) + Fh2 (P2) (EHi(pi) = \/mi + Pi2) gives info on the interaction between Hi and H2. Luscher derived a rigorous relation between the energy shift En — EH1 — EH2 and the elastic phase shift 6(s) for H1H2 scattering in continuum [1]. The measured energies En can be used to extract the phase shift 6(s) evaluated at s = En — P2, where En is the energy of the system and P its total momentum. In order to extract 6(s) at several different values of s, the simulations are done for several choices of total momenta P of the H1H2 system, which leads to different values of s = E^n — P2. • The resulting dependence of 6(s) as a function of s can be used to extract the mass mR and the width rR of the resonance R, which appears in the elastic channel H1 H2 —> R —> H1 H2. For this purpose, the 6(s) can be fitted with a Breit-Wigner form or some other phenomenologically inspired form, which depend on mR and rR. * Talk delivered by S. Prelovsek Pi ;p2 L >> few fm JP2 L= few fm Fig. 1. The energy of two hadrons in a box of size L. On the left, L ^ fm and E(L) ~ EH1 (p1) + EH2 (p2). On the right, L ~ few fm and energy gets shifted due to their interaction, i.e. E(L) ~ Ehi (p1) + Eh2(p2) + AE(L). The described method, needed for the determination of the resonance width rR, is rather challenging. It requires very accurate determination of a few lowest energy levels of the system Hi H2, since the resulting phase shift depends ultimately on the energy shift. Among all the meson resonances, this method has been up to now rigorously applied only to p resonance. Although Liischer proposed the method already in late 80's [1], the first lattice attempt to employ it to hadronic resonances had to wait until 2007 [2]. Since then, several studies of p have been carried out [3,4], with the most up to date ones [5-7]. This talk briefly describes the method to extract 6(s), mR and rR on an example of nn scattering in the p channel. It is based on a recent simulation [6], which is the statistically most accurate determination of any strong meson width on one lattice ensemble. The purpose of this talk is to highlight the main physical reasoning, which lies behind the lattice extraction of 6(s), mR and rR, omitting most of technical details. The sections follow the order of steps required, which are listed as items in the introduction. Section II describes the determination of spectrum En of the coupled system H1H2 <-> R. The Section III described why En allow one to extract the elastic phase shift 6(s). The extraction of the resonance parameters mR and rR from the phase shift 6(s) is done in Section IV. We end with conclusions. 2 Spectrum of two hadrons in a finite box The p meson is a resonance in nn scattering in P-wave, and has quantum numbers IG(JPC) = 1 + (1 ). The total momentum P of the coupled nn — p system can have values -^-d , d e Z3 due to the periodic boundary condition in the spatial direction, and we use the following three choices P= (0,0,0) , ^(0,0,1) , ^(1,1,0) and permutations . (1) This enables us to obtain several values of s = E^ — P2 for the system, thereby allowing the determination of 6(s) for these values of s without changing the spatial volume. Our simulation is performed on an ensemble of 280 [8] gauge configurations with dynamical u/d quarks, where the valence and dynamical quarks employ improved Wilson-Clover action. The corresponding pion mass is mna = 0.1673 ± 0.0016 or mn = 266 ± 4 MeV. The lattice spacing is a = 0.1239 ± 0.0013 fm and we employ a rather small volume N^ x NT = 163 x 32, which allows us to use the costly full distillation method [9] for evaluating the quark contractions. On the lattice, the discrete energies of the system En can be extracted after computing the dependence of the correlation matrix Ctj (tf,tt) on Euclidean time tf - tt Ctj(tf,tt) = (0Oi(tf) 0|(ti)|0> = ^(OiJnXnlOf> e-En(tf-ti) . (2) n The analytical expression on the right is obtained by inserting the complete set Hn ln> (nl of physical states n with given quantum numbers. The interpolators Ot have the quantum numbers of the system in question. In our case the interpolators have quantum numbers JPC = 1 and |I, I3> = |1,0> and total three-momentum P. They have to couple well to the nn state and the quark-antiquark resonance p. For each choice of P (1), we use 16 interpolators, listed in detail in Eq. (21) of [6]. We employ fifteen interpolators of quark-antiquark type Oiqq(t) = eiP" 75fa-^iU (t,x) + dfid (t,x)] , (3) x where Ft denotes different color-spin-space structures with the same resulting quantum number JPC = 1 and |I,I3> = 11, 0>. We use also one n(pi)n(p2) interpolator, where each pion is projected to a definite momentum On"(t) = ^[7T+(p1)7T-(p2)-7T-(p1)7T+(p2)] , Pl + p2 = P , n±(pi)= X etPiX qY5T±q (t, x) (4) x In practice, the nn interpolator is the most important among our 16 interpolators, since it couples to the scattering state much better than the quark-antiquark interpolators. Let us note that all other lattice studies aimed at rp used at most one quark-antiquark and one nn interpolator, which may not always allow for reliable extraction of the first excited energy level E2. Given the 16 interpolators, we compute the 16x16 correlation matrix Ctj (tf, tt) for all initial and final time-slices tt,tf = 1, ..,NT = 32. The needed Wick contractions that enter the correlation matrix with our q q and nn interpolators are depicted in Fig. 2. The contributions (a,c,e) in Fig. 2 cannot be evaluated solely from the quark propagator from one point (tt, xt) to all other points of the lattice (such a propagator allowed most of the spectroscopy studies in the past). The contributions (a,c,e) require the propagators from all and to all points on the lattice, which is too costly to evaluate in practice. We use the recently proposed distillation method for this purpose [9], which enables the exact computation of the required contractions. (a) (c) Fig. 2. Contractions for I = 1 correlators with q q (3) and nn (4) interpolators. p level n En a sa2 6 ^f (0,0,0) tM0,0,0) 1 2 0.5107(40) 0.9002(101) 0.2608(41) 0.8103(182) 130.56(1.37) 146.03 (6.58) [*] TMO,O,1) TMO,O,1) 1 2 0.5517(26) 0.6845(49) 0.1579(29) 0.3260(69) 3.06 (0.06) 156.41(1.56) ^(1,1,0) ^(1,1,0) 1 2 0.6933(33) 0.7868(116) 0.1926(49) 0.3375(191) 6.87(0.38) 164.25(3.53) Table 1. The results for two lowest levels n = 1,2 of the coupled nn — p system with three choices of total momentum P on our lattice with mna = 0.1673 ± 0.0016, L = 16a and the lattice spacing a = 0.1239 ± 0.0013 fm. The energy levels En are obtained by multiplying En a with a-1 ~ 1.6 GeV. The invariant mass squared of the system is s = E^ — P2, but the dimensionless value in the table s a2 is obtained using the discretized version of this relation [6]. We average the resulting correlators (i) over all initial time slices tt at fixed time separation tf — tt, (ii) over all directions of momenta P (1) and (iii) over all directions of the p meson polarization. The time dependence tf — tt of the correlators Ctj (tf, tt) (2) contains the information on the energies of the system En, and several methods for extracting En from Ctj are available. We extract two lowest energy levels En=1,2 of the system from the 16 x 16 correlation matrix Ctj (tf, tt) using the so called variational method [10], which is the most established among the available methods. Table 1 displays the extracted lowest two energies En=1,2 of the coupled nn — p system for our three choices of total momenta P (1). The spectrum En in Table 1 for our finite box is the main result of this section. Each energy level corresponds to a different value of s = E^ — P2, as calculated from En and P in the Table 1. In fact, the table lists values of s obtained from the discrete lattice version of the dispersion relation, which takes into account part of the corrections to s = E^ — P2 due to finite lattice spacing [6]. 3 Extraction of the phase shifts from energy levels Let us consider the case when the resonance R can strongly decay only to two spinless hadrons Hi and H2, so one has elastic scattering of Hi and H2. We point out that the non-elastic case, when a resonance can decay strongly to several final states (i.e. H1H2 and H' H2), is much more challenging for a lattice study. Suppose one encloses two hadrons H1 (p1) H2(p2) with three-momenta p1 and p2 into a large box of size L ^ fm and measures their energy. In a large box, they hardly interact and their energy is equal to sum of individual energies Enon-int = EHi(pi)+EH2(p2) with EH(p) = ^m^+p2. Now, let's force H^pO and H2(p2) to interact by decreasing the size of the box to L of a few fm. The energy of the system E(L) = EH1 (p1) + EH2(p2) + AE(L) is shifted with respect to Enon-int: it will increase (AE(L) > 0) if the interaction is repulsive and decrease (AE(L) < 0) if the interaction is attractive. This simple physical reasoning indicates that the energy shift AE(L) gives info on the interaction. Fig. 3. The scattering of two interacting particles as series of the interaction vertex M(6L) and the scattering of non-interacting particles F at finite L [11]. In fact, the energy shift AE(L) and the energy itself E(L) do not only give us "some" info on the interaction. According to the seminal analytic work of Liischer [1], E(L) or AE(L) rigorously tells us the value of the elastic scattering phase shift of H1 H2 scattering at L —> oo, i.e. 6(L = oo): Luscher method : E(L)—> 6(s,L = oo) s = E(L)2 - P2 (5) The derivation and the resulting formulae between E(L) and 6 are lengthy and rather complicated, but let us briefly explain at least why E(L) contains info on 6(L = oo). A nice and clear quantum-filed theory derivation is given in [11] and the main message is illustrated in Fig. 3. The scattering of two interacting spin-less hadrons Hi H2 at finite L (for degenerate case mHi = mH2 = m) is represented in QFT by series of: • scattering of two non-interacting hadrons at finite L, represented by F. The expression F contains sums over the loop momenta k, which are allowed in a finite box L with periodic boundary conditions in space. Here f (k0, k) stands for dependence of the vertices on the left and right on k0 and k. • the interaction vertex M with four hadron legs. This vertex depends on the elastic phase shift 6^ (at infinite volume) for the case of elastic scattering in the l-th partial wave. The physical scattering requires resummation of the bubbles in Fig. 3, with non-interacting parts F and the interacting parts M, giving AF^j^A'. The positions of the poles of the sum AF j J^A' obviously depend on M and therefore on 6i. The positions of the poles dictate the possible energy levels of the system En(L), so the energy levels En(L) depend on M and therefore on 6^. The purpose of the above illustration was just to indicate why En(L) depend on 6^. In the case of nn with JP = 1the relevant wave has l = 1 and we denote the corresponding phase by 6 = 6i. The complete analytic relations between En(L) and 6(s) needed for our case of the nn scattering with JPC = 1 and I = 1 are provided in [6] (for every |P| a different form of relation applies). These allow to extract 6 for each of our six energy levels in Table 1 and the resulting phase shifts are given in the same Table. The presented Lüscher formalism applies only for the case of elastic scattering. The nn state is the only scattering state in this channel for energies when 4n state cannot be created, i.e., when s = E^ < (4mn)2. For our mna = 0.1673 this is valid for all six levels, with the exception of the level Ei at P = 0, which is above 4n inelastic threshold. As the Lüscher analysis is not valid above the inelastic threshold, we omit this level from further analysis. The resulting scattering phase shifts for five values of s are shown in Fig. 4. This is the main result of the lattice study; the resonance properties will be obtained by fitting 6(s) in the next section. Note that the resulting phases are determined with a relatively good precision, which is better than in other available lattice studies of p at comparable u/d quark masses. The good precision can be traced back to various advanced techniques we used: the distillation method for evaluating contractions, usage of a large interpolator basis and average over all initial time slices, directions of momenta P and polarizations of p. 4 Extracting resonance mass and width from the phase shift The phase shift 6(s) in Fig. 4, obtained directly from the lattice study, can be used to extract the properties of the resonance, in our case the p. The phase shift has a typical resonance shape: it passes from 6 ~ 0° to 6 ~ 180°: the point where it crosses 90° gives the position of the resonance (s = mp), while the steepness of the rise gives its width rp. In particular, 6 is related to resonance parameters by expressing the scattering amplitude a^ in terms of 6 on one hand, and with Breit-Wigner form in the vicinity of the resonance on the other hand — a/s F(s) e2i6(s) _ 1 Q1 = -2 , • rr( ^ = -T-- • (6) s — mp + iys I (s) 2i Relation (6) can be conveniently re-written as v^Hs) cot6(s) =m2p -s . (7) co 180 I-r 16014012010080604020- 0.1 0.15 0.2 0.25 s 0.3 0.35 0.4 Fig. 4. The nn phase shift 6 (s) (in degrees) for five different values of dimensionless sa2 = (Ena)2 — (Pa)2, extracted from our lattice study [6]. The s is obtained by multiplying sa2 with (a-1)2 ~ (1.6 GeV)2. lattice (this work [6]) exp [PDG] xWn ~ 266 MeV TTVp 792 ± 12 MeV 775 MeV 9 p7t7t 5.13 ±0.20 5.97 Table 2. Our lattice results for the resonance parameters [6], compared to the experimental values. The decay width significantly depends on the phase space and therefore on mn, so the decay width extracted at mn ~ 266 MeV could not be directly compared to the measured width. So, it is customary to extract the p —> nn coupling gpnn instead of the width, where the width p*3 g2 r(s) rP = rK) (8) depends on the phase space for a P-wave decay and the coupling gpnn. The coupling is expected to be only mildly dependent on mn, which was explicitly confirmed in the lattice studies [5,7] and analytic study [12]. In (8), p* denotes the pion momentum in the center-of-momentum frame and we extract it from s using a discretized version of relation a/s = 2y/m2 + p*2 [6]. Inserting F(s) (8) into (7), one obtains an expression for 6(s) in terms of two unknown parameters: mp and gpnn. We fit these two parameters using five values of 6(s) given in Fig. 4 and Table 1, and we get the values of resonance parameters in Table 2 with small statistical errors. The resulting p-meson mass in Table 2 is slightly higher than in experiment, as expected due to mn = 266 MeV > mnxp. The coupling gpnn is rather close to the value gpnt derived from the experimental width r^xp. Fig. 5. The crosses are the nn phase shift 6(s) (in degrees) for five different values of di-mensionless sa2 = (Ena)2 — (Pa)2, extracted from our lattice study [6]. The line is the Breit-Wigner fit (7,8) for the resulting mp and gpnn in Table 2. The physical value of s is obtained by multiplying sa2 with (a-1 )2 ~ (1.6 GeV)2. 5 Comparison to other lattice and analytical studies The comparison of our results for mp and rp to two recent lattice studies [5,7] is compiled in Fig. 8 of [7]. Our result has the smallest error on a given ensemble, demonstrating that accurate lattice determination mR and rR for (some) resonances is possible now. The other two lattice studies are done for two [7] and four [5] pion masses and explicitly demonstrate mild dependence of gpnn on mn. The discussion concerning the (dis)agreement of the three lattice studies is given in [7] and will be extended in [13]. The comparison of our 6(s) to the prediction of the lowest non-trivial order of unitarized Chiral Perturbation Theory [14] is given by the solid line in 6, which has been recalculated for our mn = 266 MeV in [15]. The lowest1 order prediction does not depend on unknown LECs and agrees reasonably well with our lattice result, given by the bullets. 6 Conclusions We highlighted the main physical reasoning, which lies behind the lattice extraction of elastic phase shifts 6(s) and the resonance parameters mR and rR. The purpose was to present the general principle of the method and omit the technical details. The method was presented on the example of nn —> p —> nn scattering. This example demonstrates that a proper first-principle treatment of some hadronic resonances on the lattice is now possible. 1 One cannot make a fair comparison between out lattice result and the next-to-lowest order prediction, since it depends on a number of LECs, and some of them have been fixed using mp from another lattice study, which gets a significantly higher mp . iso 100 120 140 80 20 40 so 0 150 - -1 PI; _----1 PI; --Ur ------Ur - ............Ur 011 nn+ and p —> pn0 fluctuation in agreement with the observed value in the (e + p —> e+ forward neutron+X) experiment. 1 Introduction First we review some evidence for the role of the pion cloud in explaining nucleon observables. As examples of low-energy processes, we quote the magnetic polarizabilities [1] and electroexcitation of the nucleon [2-4]. The pion cloud acts as a coil and gives a diamagnetic contribution while the virtual excitation of the N-like quark core into the A-like quark core acts as a paramagnet. The magnetic polar-izability of the nucleon results from an approximate cancellation between these two contributions. Without the pion cloud, the paramagnetic contribution would dominate and give much too large magnetic polarizability. In the electroexcita-tion of the nucleon into A and into the Roper resonance, the linear a-model with quarks and the cloudy bag model help us understand why (40 - 50)% of the dominant M1 ampliutude and 100% of the E2 amplitude is due to pion cloud. The question arises, whether the same amplitude of the pion cloud (or equiv-alently, the same probability of pion fluctuation) can explain also observables measured at higher energies where the stucture functions of quarks play a role and pion is seen through its contribution to the corresponding quark and antiquark structure function. 2 Pion cloud in quarks can explain nucleon observables The notion of the constituent quark applies generally to the massive quark dressed by gluons, the constituent of the nucleon. This non-relativistic model with three * Talk delivered by M. Rosina massive constituent quarks works well for the hadronic masses and the magnetic moments. It breaks down if the spin properties of the baryons are considered. The improved version, the chiral constituent quark model is surprisingly suc-cesful in explaining the spin properties of nucleons and hyperons. In the simplest form applied to the nucleon the chiral constituent quark is composed of a massive quark accompanied by a quark-antiquark pair coupled to the spin-parity quantum numbers of the pion Jn = 0-. In the following we write the pion symbol as a shortcut to the quark-antiquark pair coupled to the pion quantum numbers. This simple model has been first applied by Eichten et al. [5] to explain the flavor asymmetry of the sea quarks and further elaborated by Baumgartner et al. [6] and Pirner [7] in the interpretation of the spin properties of the nucleon. It is related to the three-flavour extension proposed by Cheng and Li [8]. Explicitly written, the chiral constituent up-quark (u) structure is |u> = \J( 1 - |a) |u) - V^|d7t+) + |U7T°), (1) and of the down quark (d) Id) = - | a) Id) + v^Iutt-) - ^||d7t°). (2) The basis of pure flavour quarks is denoted by boldface u and d. At Q2 « 0 gluons do not appear as an explicit degree of freedom and the nucleon is composed of quarks and quark-antiquark pairs. Thus in the lowest order the Fock state of the constituent quark has the form (1 and 2), where in the second and third term the quark-antiquark pair is coupled to the Jn = 0- quantum nummbers of the pion. This simple structure of the chiral constituent quark (1) has two attractive features. Firstly, as we will show, the chiral constituent quark reproduces the experimental results of the deep inelastic scattering and axial-vector beta decays of the neutron quantitatively; secondly, this model complies with our picture of the origin of the quark mass by the chiral symmetry breaking mechanism of Nambu and Jona-Lasino [9]. Dressing the light quark by gluons is inevitably accompanied by creation of the Goldstone boson, the pion. The Goldstone pion is an inherent part of the constituent quark. The parameter a of (1, 2) is usually determined from the value of the axial vector coupling constant gA = 1.269 ± 0.003 [11] yielding a = 0.239 ± 0.002. The parameter a measures the probability of the constituent quark to be in the state accompanied with a charged pion. Furthermore, with the probability a/2 the constituent quark is in a state component with the neutral pion. Thus the total probability of finding a pion in the constituent quark amounts thus to slightly more than one third. The large probability of the pion in the constituent quark is best manifested in the measurements of the quark polarization in the deep inelastic scattering. Not only that one third of the constituent quark with the pion does not contribute to the spin polarization, but even more, with the oppositely oriented quarks reduces the total quark polarization to one third of what would be without the pions. The loss of the angular momentum because of the oppositely oriented quark is compensated by the orbital angular momentum of the pion in the p-state. The comparison of the experimental results of the deep inelastic scattering with the prediction of the chiral constituent-quark model is given in [10]. It is also worthwhile to mention that the valence-quark distribution does not peak at Bjorken x = 0.3 but it is softer and peaks at x = 0.2 corresponding to five and not three constitunts of the proton even before gluons can get excited. Eichten et al. ([5]) ascribe these quark-antiquark pairs to an asymmetric sea. We consider also other observables which depend strongly on the pions in the nucleon: the Gottfried sum rule IG (with corrections discussed in [10]), the integrals of the spin structure functions of proton Ip and deuteron Id and the quark spin polarization AI [12,13]. They have larger error bars than gA, but they agree reasonably well eith the model (Table 1.). The new experimental value for AI supports even more our assumption that the main contribution to the spin reduction comes from the pion fluctuation. observable model value 9a = : 1.269 ± 0.003 #C1 - a) input ig = 0.216 ± 0.033 - 2a) = 0.174 ± 0.002 ip = 0.120 ± 0.017 fsV - 2a) = 0.145 ± 0.002 Id = 0.043 ± 0.006 36 1 1 - 3a) = 0.039 ± 0.001 AI = : 0.330 ± 0.064 (1 - 3a) = 0.283 ± 0.006 Table 1. The probability a = 0.239 ± 0.002 is used to calculate different observables 3 The proton contains a neutron plus pion component Let us consider the matrix element (nn+|p). Inserting for constituent quarks our chiral quarks it is evident that the (nn+| has an overlapp with a Fock component of the proton. The result of the explicit calculation is 3 |(n7t+|p)|2 = |(d7t+|u)|2 = (1 - - a) a = 0.15. (3) The result (3) means that the constituent u quark has a component of the d quark and a pion. Although the proton has two u quarks there is no factor 2 in the amplitude, due to the flavor-spin-color structure of the nucleon. The flavor-spin wavefunction of the proton has a mixed symmetry combined into a symmetric flavor-spin function: P = 1 2 3 1 2 [Ï + V 2 S 1 3 X f 1 3 3 2 2 s (4) x f A similar expression stays for the neutron. Since the combined wavefunction is symmetric under all permutations it is enough to look at the contribution of the particles 1 and 2. In the first term of the proton wavefunction the particles 1 and 2 are symmetric and can both be u quarks and contribute constructively to the matrix element with a factor of two. In the second term the interference is destructive and the contribution cancels. Thus only the first term contributes to the matrix element. Since both in proton and in neutron the first term appears with a factor \f\Jl, the factor two is canceled out. This qualitative explanation can be verified by writing down the three-quark wavefunctions explicitly. This can be seen even easier in the isospin formalism. In the act of producing a positive pion, the corresponding u quark loses one unit of charge, it becomes a d quark. This can be described with the operator Y.11_(i) = T_ where T_ = Tx — iTy. We conveniently took the sum over all three quarks since the third quark, d, contributes zero anyway. The expectation value is < TM — 1 |T_|TM >= ^JT[T + 1) - M(M — 1) which for proton (T = 1 /2, M = 1/2) gives in fact the factor 1. It is instructive to compare with A+ (T = 3/2, M = 1 /2) in the process ep—> eAX where one gets the factor 2, pointing out that the two u quarks are always symmetric and interfere constructively. Of course, for the squared amplitude, we get the additional factor a since only the n+-dressed component of the u-quark contributes, and the factor (1 — |a) for the naked component of the final d-quark. 4 Experimental test of the pion fluctuation The pion fluctuation of nucleon is well known in the classical nuclear physics as anomalously large pion-nucleon coupling constant g2/4n = 13.6. Many of the nucleon properties are ascribed to the pion cloud of the nucleon [14]. Hovewer, there is no direct way of determinig experimentally the probability of finding a pion fluctuation in the proton. The best way is to calculate the pion flow by using the pion-nucleon coupling constant and the form factor assuming that the pion is emitted by a proton [15], [16] = "(sirtjilGWl1. (5) The pion flow is related to the measured cross section by dcY*p^nX = f„+/p(xL,t) • daY*n+^X (6) where the (y*n+ —> X) DIS cross section is assumed to be 2/3 of the (y*p —> X) DIS cross section in the cited analysis, with corrections due to absorption [10]. Obviously the pion is not emitted by a proton but by a quark. But as we showed above the state of the pion is dictated by the proton wave function and the pion form factor simulated well the assumption that the emission is from the proton. In the series of experiments [17]- [18], [19] measuring the spectrum of the forward neutrons in the reaction (e+p—>e+forward n+X) has been shown that the high energy end of the neutron spectrum is consistent with the assumption that the deep inelastic scattering takes place on the pion. Thus we are justified to say that the forward neutron is the signature of the reaction taking place on the pion and that the total probability of finding a pion in ep—mn+ fluctuation can be obtained by integrating over the variables of the pion flow. The analysis depends to some extent on the estimation of pion flux fn+ /p. The analysis has been elaborated in [10] and the quoted results are (nn+|p)2 = 0.165 ± 0.01 and 0.175 ± 0.01, respectively, for the two form factors best fitting to the experiment in [15] and [16]. 5 Conclusion The pion fluctuation p—rn+n+ and p—>p+n0 is an artifact of the quark-antiquark pairs of the constituent quarks. The impressive agreement between the measured and the calculated ratios between the probability of the pion fluctuation and the probability of finding a quark-antiquark pair of the constituent quark is a strong support of the constituent quark model. In this section we stress the difference between the notion of the quarkantiquark pairs coupled to the pion quantum numbers being part of the constituent quarks and the pions of the proton. While the quark-antiquark pairs are implied by the experimental values of gA, the integrated spin structure functions and the violation of the Gottfried summ rule, the fluctuating pions are identified by the characteristic energy and pT distribution of the neutron spectra in the ep —> nn+ reaction. Eichten et al. [5] have named the quark-antiquark pairs of the constituent quark the asymmetric quark sea. This name emphasizes hopefully sufficiently the difference of their origin as compared to the normal quark sea. For the value a = (dn+|u)2 = 0.24 each quark contains 0.36 quark-antiquark pairs. Summing up the quark-antiquark pairs one obtains about one quark-antiquark pair per nucleon. Using this value of a gives (nn+|p)2 = 0.15. This number corresponds well with the experimental value of (nn+|p)2 = 0.165 ± 0.01 or 0.175 ± 0.01. It follows that in « 0.26 cases the proton is a neutron+ or a proton+n0. This means that about one quater of the nucleon's quark-antiquark pairs show up as the pion fluctuation. Acknowledgments We wish to thank A. Bunyatyan for the discussions of the analysis on the forward neutron spectra and K. Rith for pointing us out the new results on spin-polarization data in DIS. References 1. B. Golli and R. Sraka, Phys. Lett. B 312 (1993) 24-29. 2. M. Fiolhais, B. Golli and S. Sirca, Phys. Lett. B 373 (1996) 229-234. 3. B. Golli and S. Sirca, Eur. Phys. J. A 38 (2008) 271-286. 4. B. Golli, S. Sirca and M. Fiolhais, Pion electro-production in the Roper region in chiral quark models, Eur. Phys. J. A 42 (2009) 185-193. 5. E.J.Eichten, I.Hinchliffe and C.Quigg, Phys. Rev. D 45 (1992) 2269. 6. S. Baumgaertner, H. J. Pirner, K. Koenigsmann, B. Povh, Z. Phys. A 353 (1996) 397. 7. H. J. Pirner, Prog. Part. Nucl. Phys. 36 (1996) 19-28. 8. T. P. Cheng and Ling-Fong Li, Phys. Rev. Lett. 74 (1995) 2872-2875. 9. S.P. Klevansky, Rev. Mod. Phys. 64 (1992) 694. 10. A. Bunyatyan and B. Povh, Eur. Phys. J. A 27 2006 359-364. 11. K. Nakamura et al. (Particle Data Group), J. Phys. G 37 (2010) 075021. 12. K. Rith, Prog. Part. Nucl. Phys. 49 (2002) 245. 13. A. Airapetian et al., Phys. Rev. D 75 (2007), 012007. 14. A. W. Thomas, Prog. Theor. Phys. 168 (2007) 614 15. B. Kopeliovich, B. Povh and I. Potashnikova, Z. Phys. C 73 (1996) 125, [hep-ph/9601291]. 16. H. Holtmann et al, Phys. Lett. B 338 (1994) 363. 17. C. Adloff et al. [H1 Collaboration], Eur. Phys. J. C 6 (1999) 587, [hep-ex/9811013]. 18. A. Aktas et al. [H1 Collaboration], Eur. Phys. J. C 41 (2005) 273, [hep-ex/0501074]. 19. S. Chekanov et al. [ZEUS Collaboration], Phys. Lett. B 610 (2005) 199, [hep-ex/0404002], and references therein.