Diquarks and higher twist effects: recent results 1. Francisco Caruso

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1 Diquarks and higher twist effects: recent results 1 Francisco Caruso Centro Brasileiro de Pesquisas Físicas/CNPq Rua Dr. Xavier Sigaud 150, , Rio de Janeiro, Brazil Instituto de Física da Universidade do Estado do Rio de Janeiro R. São Francisco Xavier 54, , Rio de Janeiro, Brazil Em homenagem ao Prof. Paulo Leal Ferreira ABSTRACT A quark-diquark picture of the nucleon, previously introduced in the description of several exclusive and inclusive processes at intermediate Q values, is found to model the higher twist data on the unpolarized nucleon structure function F (x, Q ); the main results are summarized. The emerging set of parameters is consistent with the diquark properties suggested by other experimental and theoretical analyses. Within this picture, higher twist effects is expected to be measured in deep inelastic processes induced by neutrinos. Higher twist corrections to the Bjorken sum rule are also estimated in the framework of this quark-diquark model of the nucleon. The resulting corrections to the Bjorken sum rule turn out to be negligible. 1. Introduction Recent data from Deep Inelastic Scattering (DIS) experiments at CERN and SLAC (Fig. 1) have provided precise information on the unpolarized nucleon structure function F (x, Q ) and have allowed a quantitative estimate of higher twist terms for protons [1,], parameterized as ( F (x, Q ) = F LT (x, Q ) 1 + C(x) ) Q (1) where F is the measured structure function, F LT is the leading twist contribution and F LT C(x)/Q is the higher twist term. 1 This paper reports part of a wide research program on diquarks in collaboration with M. Anselmino, V. Barone, E. Cheb-Terrab, J.R.T. de Mello Neto, E. Leader, E. Levin, F. Murgia, A. Penna Firme, E. Predazzi, P.C. Quintairos and J. Soares. 1

2 The dashed curves in Fig. 1 show the Q dependence of F LT for different values of x, while the solid curves are the fitting including higher twist contributions. Figure 1: The peculiar property of this parametrization is that the function C(x) changes sign, being negative at small x values and positive at larger ones, as it is shown in Fig.. It may be stressed that the addition of a D(x)/Q 4 term in Eq. (1) would equally well reproduce the data [1]. Higher twist contributions to DIS are expected to originate from quark and gluon correlations, and it is assumed that they can be described by a quark diquark picture of the nucleon. Diquarks originate from QCD color forces and model correlations between two quarks inside the nucleon which, at moderate Q values, interact collectively and behave as bound states. Evidence in support of the diquark model is given by a number of phenomenological successes in the description of several physical phenomena, such as: the behavior of the DIS structure functions at Bjorken x 1, the inclusive large p T deuteron production observed in 70 GeV/c pp interaction at Serpukhov, the large p T baryon production in pp interactions,

3 Figure : the observed (but scarce) production of ++ at large transverse momentum, some exclusive processes [3] and many other processes reviewed in Ref. [4]. Therefore, there is little doubt that two quark correlations are present inside nucleons and that they might play a significant rôle in some processes at moderate Q values, precisely the region where the higher twist effects have been observed (Q up to 10 0 GeV ).. Diquark contributions to F The diquark contributions to DIS have been studied in several papers [5,6], mainly taking into account only spin 0, scalar diquarks, in simplified versions of the quark-diquark model of the nucleon. A most general calculation of spin 0 and spin 1 diquark contributions to DIS was performed in Ref. [7], allowing for a vector diquark anomalous magnetic moment and for scalar-vector and vector-scalar diquark transitions; however, at that time, no attempt was made of fitting the experimental data due to the lack of detailed information on the higher twist contributions alone. This work was done recently in Ref. [8]. Let us assume the nucleon to be a quark diquark state where the diquark models correlations between two quarks which, at moderate momentum transfers, might interact collectively and behave as a bound state. When probing the nucleon with the virtual photon in DIS we have then three kinds of contributions: the scattering off the single quark, the elastic scattering off 3

4 the diquark, and the inelastic diquark contribution, that is the scattering off one of the quarks inside the diquark. Eventually, at large enough Q values, the elastic diquark contribution, weighted by form factors, vanishes and one recovers the usual pure quark results. The expression of the structure functions F, in the quark-diquark parton model, is then given in general by: F (x, Q ) = q F (q) + S F (S) + V F (V ) + + q S F (q S ) + q V F (q V ) + S,V F (S V ) + S,V F (V S) () where (q) denotes the single quark contribution, (S) and (V ) respectively the scalar and vector diquark ones, and (q S ) ((q V )) the contribution of the quark inside the scalar (vector) diquark. We have also allowed for elastic diquark contributions with a scalar-vector (S V ) or vector-scalar (V S) transition. Let us consider here the unpolarized structure function F (x, Q ) only. It is straightforward to extract from the full contribution of the quark-diquark nucleon model to F all terms which are proportional to the diquark form factors [8]. These terms vanish at large Q values, but at moderate Q values they give non negligible contributions. These are the terms which we assume to model higher twist effects in DIS. Explicitly following the notations of Ref. [7,8] they are given by F HT = S e S S(x) x D S + V {[( 1 3 e V V (x) x 1 + ν ) D 1 + m N x ν m N x D + m N νx ( 1 + ν ) ] [ D 3 + D1 + m N x + 1 e S S(x) x m N ν DT + 1 e S V (x) x m N ν DT + 6 S V ν ] } m N x D + (3) q S e q S x q S (x, Q )D S q V e q V x q V (x, Q )D V Notice that, depending on the actual behavior of the form factors, one not only obtains contributions proportional to 1/Q, but also to higher powers of 1/Q, which might be significant in the lowest Q range of the experimental data [1,]. Notice also that Eq. (3) has both positive and negative contributions, so that it might yield positive or negative higher twist values, according to the different x or Q ranges. 4

5 The valence quark content of the proton in the quark-diquark model [9] is given by the flavour and spin wave function: p, S z = ±1/ >= ± 1 3 { sin Ω [ V ±1 (ud) u V ±1 (uu) d + + V 0 (uu) d± V 0 (ud) u±] 3 cos Ω S (ud) u ±} (4) where V ±1 (ud) stays for a (ud) vector diquark with the third component of the spin S z = ±1, u is a u quark with S z = 1/ and so on. The vector and scalar diquark components have different weights, so that the probabilities of finding a vector or scalar diquark in the proton are sin Ω and cos Ω respectively. From Eqs. (3,4) and folowing the notation of Ref. [8] we finally get, for a proton (F HT ) p = 1 [ ] 9 cos Ω f S (x) [4f us (x, Q ) + f ds (x, Q )] x DS+ {[( ) sin Ω [f V(ud) (x) + 3f V(uu) (x)] x 1 + ν m N x D 1 + ( ) ] [ ]} ν m N x D + m N νx 1 + ν D 3 + D1 + ν m N x m N x D + (5) 1 7 sin Ω [16f uv(uu) (x, Q ) + 5f uv(ud) (x, Q )] x D V [ cos Ω f S (x) sin Ω f V(ud) (x) ] x m N ν DT We have adopted the following parametrization of the different distribution functions appearing in Eq. (5): f P (x) = N P x α P (1 x) β P (6) for each kind of parton P (P = S, V, q S, q V ) and the N P s are the proper normalization constants. Concerning the diquark form factors we have chosen the most simple expressions which agree 5

6 both with the asymptotic perturbative QCD predictions [10] and the point-like, Q 0, limits: D S = Q S Q S + Q ( Q ) D 1 = V Q V + Q D = (1 + κ)d 1 (7) (7) D 3 = Q m 4 D1 N D V = D 1 Q D T = m N D 1 where κ is the vector diquark anomalous magnetic moment. We have then used Eq. (5), with the parameterizations given in Eqs. (6) and (7), as a model to fit the experimental data on the higher twist contributions to F p [1,11]. We have neglected the small effect due to the perturbative QCD Q evolution of the single quark distribution functions f us, f ds, f uv(ud) and f uv(uu). The results of our best fits [8], using the MINUIT program (version 9.1) of CERNLIB, are indeed very good. Fig. 3 shows F HT for three values of x, namely x = 0.07, 0.18, Results for other x values are given in Ref. [8]. The corresponding values of the free parameters turn out to be: cos Ω = 0.81 Q S = 1.4 GeV Q V = 1.10 GeV α S =.13 β S = α V = 7.93 β V = 3.3 (8) β us = 5.13 β ds = 5.13 β uv(ud) = 8.41 β uv(uu) = 8.41 The values of α us, α ds, α uv(ud) and α uv(uu) have been fixed to be 0.5 in agreement with the expected small x behaviour of the valence quark distributions. The value of κ which allows the best fits is κ = 0. It is important to stress that the diquark features, as emerging from the values of the best fit parameters, Eq. (8), are consistent with the expectations from many other studies and applications of quark-diquark models of the nucleon [4]. The scalar diquarks turn out to be more abundant (cos Ω = 0.81) and more point-like (Q S > Q V ) than the vector ones. Also, the average x carried by a scalar diquark, proportional to its average mass, is smaller, as expected, 6

7 Figure 3: than the average x of a vector diquark. In Fig. 4 we plot the functions xf P (x), where the f P (x) are the quark and diquark distributions given in Eqs. (6) and (8). The total nucleon momentum fraction carried by the diquarks amounts to 0.4. The deep inelastic scattering of neutrinos on unpolarized nucleons is considered in the framework of the same (and the most general) quark-diquark model applied to the electronnucleon DIS [1]. The resulting scaling violation is discussed, but a detailed study of the higher twist terms and their possible evaluation are at present impossible, due to lack of experimental data. 3. Diquark contribution to Bjorken sum rule Several recent measurements of the polarized structure functions g p 1 (x) and gn 1 (x) [13] both for protons and neutrons have allowed the first tests of the fundamental Bjorken sum rule [14]: 1 0 dx [g p 1 (x) gn 1 (x)] = S Bj = a 3 6 E NS(Q ) (9) where a 3 = ± is the neutron β-decay axial coupling. The above equation holds on ground of isospin invariance, at leading twist in the Operator Product Expansion; 7

8 Figure 4: the coefficient function E NS has been computed up to third order [15] in the strong coupling constant α s : E NS (Q ) = 1 α ( s π 3.58 αs ) ( αs ) 3 0. (10) π π where a number of three active flavors is assumed. Data are available at Q = and 4.6 (GeV/c) [13,16,17] for the L.H.S. of Eq. (9): 0.15±0.014±0.01 and 0.01±0.039±0.050, respectively. At such values of Q a comparison of Bjorken sum rule with experiment should also take into account possible higher twist or target mass corrections, not included in Eq. (9): 1 0 dx [g p 1 (x) gn 1 (x)] = S Bj (Q ) + δs Bj HT + δsbj T (11) Some evaluations of target mass [18-0] and higher twist corrections can be found in the literature [13], either in the framework of the Drell-Hearn-Gerasimov sum rule [1] or of the QCD sum rules [18,]. Some of these corrections may be sizeable at Q (GeV/c), but the final comparison with data shows no significant violation of the Bjorken sum rule. Let us notice that reliable estimates of the higher twist contributions are of fundamental importance in view of the recent suggestions [19,3] to exploit the Bjorken sum rule (9) in order to extract the value of the strong coupling constant α s. 8

9 To this purpose another possible evaluation of higher twist corrections to S Bj in the framework of the quark-diquark model of the nucleon discussed in the previous sections was considered in Ref. [4]. We would like to stress that the diquark approach, being a physically motivated self consistent model rather than an expansion in powers of 1/Q, has the unique feature of taking into account a full set of higher twist corrections and not only the leading ones. Confident in its physical content, we adopt here the same diquark model, with the same parameters, as given in Eq. (8). Therefore the results presented here are parameter free predictions. The elastic diquark contributions have been studied in Ref. structure function g 1 (x, Q ), they read g (V ) 1 (x, Q ) = 1 4 e V V (x) [( + Q m N x [7] and, for the polarized ) (D 1 D + 1 ] Q D D 3 ) Q 4m N x D where D 1,,3 are form factors appearing in the most general coupling of the photon to a spin 1 diquark; scalar diquarks obviously do not contribute to the polarized structure functions. V denotes the difference between the number density of vector diquarks with spin parallel and antiparallel to the nucleon spin. Following the notations of Ref. [7] the higher twist diquark contributions to g 1 for protons and neutrons are given by: [ ] g HT 1 p = 1 4 ( 16 9 V uu V ud ) [( + Q m N x ] Q 4m N x D 1 ( 4 [ ] g HT 1 n = 1 ( V uu + 1 ) 9 V ud [( + Q 9 u Vuu u V ud d V ud m N x ) (D 1 D + 1 Q D D 3 ) ) D V ) (D 1 D + 1 Q D D 3 ) ] Q 4m N x D 1 ( 1 9 u + 4 Vuu 9 u V ud + 1 ) 9 d V ud DV where the suffices to V indicate the quark content of the vector diquark; the negative contributions come from the scattering off the quarks inside the diquarks, weighted by a factor (1 D V ), with D V related to the diquark form factor. In Eq. (13) we have already used isospin relationships and all distribution functions refer to the proton; q is the usual helicity density carried by quark q and the suffices refer to the type of diquark it comes from. Subtracting Eqs. (14) from (13) yields [ ] g HT 1 p n = 5 1 V uu [( + Q 1 6 u Vuu D V m N x ) (D 1 D + 1 ] Q D D 3 ) Q 4m N x D (1) (13) (14) (15) 9

10 where, V uu (x) = 4 9 sin Ω f Vuu (x) u Vuu (x) = 8 9 sin Ω f uvuu (x) (16) and the f P distributions are normalized to 1 0 dx f P (x) = 1. From Eqs. (15), (16) and (9), upon integration over x and taking, as in Ref. [8], D 1 = D = D V, we obtain [ ] δs Bj HT = sin Ω { 6D1 + 5Q D 1 D 3 diquarks Q 4m (D1 + Q dx } (17) D 1 D 3 ) N 0 x f V uu (x) Eq. (17) gives the quark-diquark model higher twist contributions to Bjorken sum rule. We take the explicit expressions of the distribution function f Vuu for a vector diquark and of the form factors from our previous application of the same model to a fit of higher twist contributions to the unpolarized structure function F (x, Q ) [8]: sin Ω = 0.19 f Vuu = Nx 7.93 (1 x) 3.3 D 1 = D = D V = ( 1.1 ) Q (18) D 3 = Q m 4 D1 N where N = 1/B(7.93; 3.3) is the normalization constant (B is the Euler beta function). From Eqs. (17) and (18) we obtain the numerical estimates at the Q values [in (GeV/c) ] for which data are available: [ ] δs Bj HT (Q =.0) diquarks [ ] δs Bj HT (Q = 4.6) diquarks The prediction of the model in the full range 1 Q 10 (GeV/c) is shown in Fig 5, which should be compared with the experimental values of S Bj reported after Eq. (10). In the Fig. 5 we compare the results given in Eqs. (19) also with the perturbative QCD higher order corrections of Eqs. (9) and (10), namely with a 3 [E NS (Q ) 1]/6. This definitely shows that the higher twist contributions to the Bjorken sum rule, evaluated in the framework (19) 10

11 of the quark-diquark model of the nucleon, turn out to be quite small. Indeed our model yields a contribution which has an opposite sign with respect to the pqcd corrections, but much smaller in magnitude: at Q = and 4.6 (GeV/c) we have, respectively, [ ] δs Bj HT (Q ) diquarks a 3 [1 E NS (Q )]/6 7% and %. (0) Figure 5: It should be stressed that the diquark model predicts a different sign for the high twist contribution to S Bj than the QCD sum rules estimates [13], while the absolute values are compatible within the errors. It is worthwhile mentioning that the negative sign of the higher twist contribution is a common feature of all estimates based on the QCD sum rule approach [13,19]. Therefore, the positiveness of our result shows that the errors for the higher twist contributions could be bigger than used to be quoted. The range of prediction for the absolute value of S Bj HT, taken from Ref. [13], is given in Fig

12 4. Why, in this case, diquark contribution is so small? We can argue that the smallness of the diquark contributions to the polarized structure functions, Eq. (0), is due to some intrinsic features of the quark-diquark model of the nucleon: the strong SU(6) violation (sin Ω = 0.19) favoring scalar diquarks; the mass scale of the vector diquark form factor which turns out to be small, Q V = 1.1 (GeV/c), corresponding to a large size, and the vector diquark x distribution which is found to be peaked at x 0.7, suggesting that vector diquarks consist of two almost uncorrelated quarks. This conclusion can be achieved by computing also the diquark contribution to the first moment of g p 1 (x) alone, i.e., Γp 1 = 1 0 dx gp 1 (x). A sizeable diquark contribution to Γp 1 would indicate that the tiny results obtained above for the Bjorken sum, Eqs. (19), are only due to a strong cancellation between the proton and the neutron contributions. However, an explicit calculation shows that diquark contributions to Γ p 1 are comparable to those found for the Bjorken sum rule, rejecting this possibility. 5. Conclusions This paper summarizes recent studies on how some observed higher twist effects could be described in terms of a quark-diquark model for the nucleon. All the work reviewed here has certainly contributed to the development of a general and physically motivated phenomenological framework, within wich it is possible to through some light on higher twist effects in electronand neutrino-initiated deep inelastic processes. In particular, the diquark contributions to the Bjorken sum rules turns out to be quite small. We hope that our estimates will help in making the use of the Bjorken sum rule in the experimental measurement of the strong coupling constant more reliable. Acknowledgements - Ao Prof. Paulo Leal Ferreira, gostaria de expressar minha mais sincera admiração e gratidão por sua incansável contribuição ao desenvolvimento da Física e da Ciência no Brasil. I would like to thank the organizers of this Symposium for the kind invitation and for their warm hospitality. I am grateful to the CBPF and the CNPq of Brazil for financial support. 1

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