- Email: [email protected]
Power corrections to the structure functions of proton and deuteron
A ELSEVIER
Nuclear Physics A666&667 (2000) 165c-168c
www.clsevier.nl/Iocate/npe
Power corrections to the structure functions of proton and deuteron G. Ricco ~ and S. Simuta b ~Dipartimento di Fisica, Universit/t di Genova and INFN - Genova, Via Dodecanneso 33, 1-16146, Genova, Italy Iqstituto Nazionale di Fisica Nucleare, Sezione Roma III, Via. della Vasca Navale 84, 1-00146 Rorna, Italy Power corrections to the Q2 behaviour of the low-order moments of both the longitudinal and transverse structure functions of proton and deuteron have been investigated using available phenomenological fits of existing data in the Q2 range between 1 and 20 (GeV/c) 2. The contributions of (target-dependent) multiparton correlations to both I/Q ~ and 1/Q 4 power terms have been determined in the transverse channel, while the longitudinal data appear to be consistent with a pure (target-independent) renormalon picture. Finally, the extracted twist-2 contribution is found to be compatible with the hypothesis of an enhanced d-quark parton distribution at large x.
1. I N T R O D U C T I O N
in the past few years various analyses of the so-called power-suppressed terms in the world data oil both the transverse F.~'(x, Q2) and longitudinal F ~ ( x , Q~) structure functions of proton and deuteron have been carried out. They have been based either on the choice of a phenomenological ansiitz [1] or on renormalon models [2], adopting for the leading twist the leading-order (LO) or the next-to-leading-order (NLO) approximations. Very recently, the effects of the next-to-next-to-leading order (NNLO) have been investigated on E~(x, Q2) and the longitudinal to transverse cross section ratio Rc/y(x , x . Q2) [3] as well as on the parity-violating structure function xEJ(x, Q2) [4]. These analyses at NNLO seem to indicate that power corrections in F~(x, Q2), t~/7(x, (22) and :rF.J(x, Q2) can be quite small; however, in such analyses the highest value of x is typically limited at x _~ 0.75 and also the adopted Q2-range is limited (Q2 >~ 5 + 6 (GeV/c) 2 at x -~ 0.75) in order to avoid the nucleon-resonance region. In this contribution we present the main results of a new analysis of the world data performed in Ref. [5], where the Q2-range of tile analysis has been extended down to Q2 ~ 1 (GeV/c) 2. In this way the sensitivity to the presence of dynamical higher twists (i.e., multiparton correlations) has been enhanced, thanks to the inclusion of the contributions of the nucleon-resonance regions and the nucleon elastic peak. Such an inclusion is clearly worthwhile also because of parton-hadron duality arguments (see, e.g., Ref. [6]). Our analysis [5] has been carried out in terms of few low-order moments of the s t r u c 0375°9474/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII S0375-9474(00)00023-3
166c
G. Ricco, S. Simula /Nuclear Physics A666&667 (2000) 165c-168c
ture functions and, therefore, in order to disentangle properly target-mass and dynamical higher-twist effects in the data, the Nachtmann definition of the moments [7] has been adopted. For the evaluation of the latter systematic measurements of the experimental structure functions are required in the whole x-range at fixed values of Q2. Since such measurements are not always available, we have adopted interpolation formulae ("pseudodata"), which fit the considerable amount of existing proton and deuteron data, covering sufficiently well the range of x crucial for the evaluation of the low-order moments considered in our analysis (see Ref. [5] for more details). 2. A N A L Y S I S
OF THE DATA
The leading twist has been treated at NLO in the strong coupling constant (~s(Q2) (see Ref. [5] for more details). Moreover, in order to try to disentangle the target-dependent multiparton correlation effects from the target-independent large-order perturbative effects we have considered explicitly the power-like terms associated to the infrared (IR) renormalon model of Ref. [2], which adopts the naive non-abelianization ( N N A ) approximation containing both 1/Q 2 and 1/Q 4 power-like terms. First of all we have checked whether the power corrections to the NLO twist2 contribution may be explained by pure IR-renormalon terms in the whole range 1 ~ Q2 (GeV/c)2 <~ 20. It turns out that a simultaneous fit of the transverse and longitudinal data is not possible, as already noted in Ref. [2]. Nevertheless, if the analysis is limited to the longitudinal channel, a good reproduction of the data can be obtained using only the power-like terms associated to the I R renormalons. Moreover, the obtained values of the [ R strengths are not inconsistent with the N N A expectations (see Ref. [5]) and nicely agree with the corresponding findings of Ref. [4], obtained from a NLO analysis of the neutrino data on xFN(x, Q2). After having determined the strengths and signs of the twist-4 and twist-6 IRrenormalon contributions from the analysis of the longitudinal channel, we have analysed the transverse data adopting the following twist expansion
M ~r ( Q ~) = # ~T( Q 2) + # ~TUn) ( Q:) + a(4)[a~(Q2)] [a~(#2)j ~(~41 ~#2+
#4 a~ ) [c~s(Q2)]~ [ ~ ( # : ) j (~) Q--7
(1)
where #~T ( Q2) is the leading twist term at NLO, #T(I,)(Q2) is the IR-renormalon contribution (fixed by the analysis of the longitudinal channel), while the logarithmic pQCD evolution of the twist-4 (twist-6) multiparton correlation contribution is accounted for by the term [a,(Q2)/a~(#2)] "~') ([as(Q2)/c~(#:)] "~6)) with an effective anomalous dimension 7~4) (3~6)) and the parameter a 4 (a~) represents the overall strength of the twist-4 (twist-6) term at the renormalization scale Q2 =/~2 (for the latter the value # = 1 GeV/c has been adopted). Some of the results based oil Eq. (1) are shown in Fig. 1. Our main results [5] can be summarised as follows: i) the higher-twist contributions to the second moment MT(Q 2) turn out to be very small in the whole Q2-range of the analysis; ii) for n >_ 4 the twist-4 and twist-6 contributions have opposite signs (see Ref. [5]), so that the total higher-twist contribution becomes smaller than its individual terms; in particular, at large Q2 the sum of the twist-4 and twist-6 contributions is just a small fraction of the twist-2 term (G 10% for Q2 ~ n (GcV/c)2); iii) for n >_ 4, the
G. Ricco, S. Simula /Nuclear Physics A666&667 (2000) 165c-168c
167c
IR-renormalon contribution increases significantly around Q2 ~ 1 (GeV/c) 2 becoming of the same order of magnitude of the twist-2 term at N L O in case of higher order moments; the effects from multiparton correlations appear to exceed the IR-renormalon term only for Q2 > 2 (GeV/c) 2 (at n _> 4). Finally, it has been checked that our determination of the multiparton correlation effects is only marginally affected by the specific value adopted for o~(M}) (see Ref. [5]).
. . . . . . . .
0.016
E
0.20
i
.
.
.
.
.
.
.
.
i
0.012
"o
0.15
0.008 0.10 0.004
0.05
0.000
0.00
10
10
Q2 (GeV/c)2
Q2 (GeV/c)2
Figure 1. Twist analysis of the proton transverse moments MT(Q 2) for n = 2 (a) and n = 6 (b). The solid lines are the results of Eq. (1) fitted by the least-x 2 procedure to the pseudo-data (open squares). The dashed lines are the twist-2 contribution, while the dotted and dot-dashed lines correspond to the IR-renormalon and multiparton correlation contributions, respectively.
An interesting feature of our analysis of the transverse moments is that the leadingtwist contribution is extracted from the data and not fixed by calculations based on a particular set of parton distributions. The comparison of our extracted twist-2 term with the predictions based on the GI~V parton distributions [8] is shown in Fig. 2 for Q2 >~ 5 (GeV/c) 2 and n = 6. It can clearly be seen that our results and the GI{V predictions agree quite well in case of the proton, whereas they differ significantly in case of the deuteron. The inclusion of a new empirical determination of the nuclear effects in the deuteron (see Ref. [3] for more details), increases only a little bit the disagreement (see dashed lines in Fig. 2). A possible solution, advocated in Ref. [3], is an enhancement of the d-quark parton distribution at large x. The resulting modified G R V predictions, including also the empirical nuclear corrections in case of the deuteron, are shown by the solid lines in Fig. 2 and agree quite well with our results. Therefore, our extracted twist-2 moments are clearly compatible with the hypothesis of an enhancement of the d-quark distribution at large x. 3. C O N C L U S I O N S The power corrections to the Q2 behaviour of the low-order moments of both the longitudinal and transverse structure functions of proton and deuteron have been analysed using available phenomenological fits of existing data in the Q2 range between 1 and 20 (Ge~Jc) 2. The effects of both target-dependent higher-twists (i.e., multiparton cor relations) and target-independent power corrections (i.e., renormalon ambiguities) have
G. Ricco, ,7. Simula /Nuclear Physics A666&667 (2000) 165c-168c
168c
O.OOS
0.004
. . . .
,
. . . .
,
(a)
. . . .
(b) 0.003
0.004
0.002
0.003 10
15
Q2 (GeV/c)2
5
10
15
Q2 (GeV/c)2
Figure 2. The leading-twist moment #T(Q2) versus Q2 for n = 6. Open dots: twist-2 extracted fi'om our analysis of the proton (a) and deuteron (b) transverse pseudo-data; the errors represent the uncertainty of the fitting procedure corresponding to one-unit increment of the x 2 / N variable. Dot-dashed lines: GRV prediction at NLO [8]. Dashed lines: the same as the dot-dashed lines, but including the empirical nuclear correction as in Ref. [3] in case of the deuteron. Solid lines: the same as the dashed lines, but including the enhancement of the d-quark distribution at large x of Ref. [3].
been determined from the data. The renormalon contribution is able to reproduce the longitudinal channel and its determination is in nice agreement with the results of a recent N L O analysis [4] of neutrino data on x F ~ ( x , (22). The contributions of multiparton correlations to both the twist-4 and twist-6 terms have been phenomenologically determined in the transverse channel and found to have non-negligible strengths with opposite signs. An interesting outcome of our analysis is that the extracted twist-2 term appears to be compatible with an enhancement of the d-quark parton distribution at large x. REFERENCES 1. M. Virchaux and A. Milsztajn: Phys. Lett. B274 (1992) 221. X. Ji and P. Unrau: Phys. Rev. D52 (1995) 72. 2. M. Dasgupta and B.R. Webber: Phys. Lett. B382 (1996) 273. E. Stein et al.: Phys. Lett. B376 (1996) 177. M. Maul et al.: Phys. Lett. B401 (1997) 100. 3. U.K. Yang and A. Bodek: Phys. Rev. Lett. 82 (1999) 2467. 4. A.V. Sidorov: these proceedings and references therein quoted. 5. G. Ricco, S. Simula and M. Battaglieri: Nucl. Phys. B555 (1999) 306. 6. G. Ricco et al.: Phys. Rev. C57 (1998) 356; Few-Body Syst. Suppl. 10 (1999) 423. 7. O. Nachtmann: Nucl. Phys. B63 (1973) 237. 8. M. Gliick, E. Reya and A. Vogt: Z. Phys. C67 (1995) 433.
Nuclear Physics A666&667 (2000) 165c-168c
www.clsevier.nl/Iocate/npe
Power corrections to the structure functions of proton and deuteron G. Ricco ~ and S. Simuta b ~Dipartimento di Fisica, Universit/t di Genova and INFN - Genova, Via Dodecanneso 33, 1-16146, Genova, Italy Iqstituto Nazionale di Fisica Nucleare, Sezione Roma III, Via. della Vasca Navale 84, 1-00146 Rorna, Italy Power corrections to the Q2 behaviour of the low-order moments of both the longitudinal and transverse structure functions of proton and deuteron have been investigated using available phenomenological fits of existing data in the Q2 range between 1 and 20 (GeV/c) 2. The contributions of (target-dependent) multiparton correlations to both I/Q ~ and 1/Q 4 power terms have been determined in the transverse channel, while the longitudinal data appear to be consistent with a pure (target-independent) renormalon picture. Finally, the extracted twist-2 contribution is found to be compatible with the hypothesis of an enhanced d-quark parton distribution at large x.
1. I N T R O D U C T I O N
in the past few years various analyses of the so-called power-suppressed terms in the world data oil both the transverse F.~'(x, Q2) and longitudinal F ~ ( x , Q~) structure functions of proton and deuteron have been carried out. They have been based either on the choice of a phenomenological ansiitz [1] or on renormalon models [2], adopting for the leading twist the leading-order (LO) or the next-to-leading-order (NLO) approximations. Very recently, the effects of the next-to-next-to-leading order (NNLO) have been investigated on E~(x, Q2) and the longitudinal to transverse cross section ratio Rc/y(x , x . Q2) [3] as well as on the parity-violating structure function xEJ(x, Q2) [4]. These analyses at NNLO seem to indicate that power corrections in F~(x, Q2), t~/7(x, (22) and :rF.J(x, Q2) can be quite small; however, in such analyses the highest value of x is typically limited at x _~ 0.75 and also the adopted Q2-range is limited (Q2 >~ 5 + 6 (GeV/c) 2 at x -~ 0.75) in order to avoid the nucleon-resonance region. In this contribution we present the main results of a new analysis of the world data performed in Ref. [5], where the Q2-range of tile analysis has been extended down to Q2 ~ 1 (GeV/c) 2. In this way the sensitivity to the presence of dynamical higher twists (i.e., multiparton correlations) has been enhanced, thanks to the inclusion of the contributions of the nucleon-resonance regions and the nucleon elastic peak. Such an inclusion is clearly worthwhile also because of parton-hadron duality arguments (see, e.g., Ref. [6]). Our analysis [5] has been carried out in terms of few low-order moments of the s t r u c 0375°9474/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII S0375-9474(00)00023-3
166c
G. Ricco, S. Simula /Nuclear Physics A666&667 (2000) 165c-168c
ture functions and, therefore, in order to disentangle properly target-mass and dynamical higher-twist effects in the data, the Nachtmann definition of the moments [7] has been adopted. For the evaluation of the latter systematic measurements of the experimental structure functions are required in the whole x-range at fixed values of Q2. Since such measurements are not always available, we have adopted interpolation formulae ("pseudodata"), which fit the considerable amount of existing proton and deuteron data, covering sufficiently well the range of x crucial for the evaluation of the low-order moments considered in our analysis (see Ref. [5] for more details). 2. A N A L Y S I S
OF THE DATA
The leading twist has been treated at NLO in the strong coupling constant (~s(Q2) (see Ref. [5] for more details). Moreover, in order to try to disentangle the target-dependent multiparton correlation effects from the target-independent large-order perturbative effects we have considered explicitly the power-like terms associated to the infrared (IR) renormalon model of Ref. [2], which adopts the naive non-abelianization ( N N A ) approximation containing both 1/Q 2 and 1/Q 4 power-like terms. First of all we have checked whether the power corrections to the NLO twist2 contribution may be explained by pure IR-renormalon terms in the whole range 1 ~ Q2 (GeV/c)2 <~ 20. It turns out that a simultaneous fit of the transverse and longitudinal data is not possible, as already noted in Ref. [2]. Nevertheless, if the analysis is limited to the longitudinal channel, a good reproduction of the data can be obtained using only the power-like terms associated to the I R renormalons. Moreover, the obtained values of the [ R strengths are not inconsistent with the N N A expectations (see Ref. [5]) and nicely agree with the corresponding findings of Ref. [4], obtained from a NLO analysis of the neutrino data on xFN(x, Q2). After having determined the strengths and signs of the twist-4 and twist-6 IRrenormalon contributions from the analysis of the longitudinal channel, we have analysed the transverse data adopting the following twist expansion
M ~r ( Q ~) = # ~T( Q 2) + # ~TUn) ( Q:) + a(4)[a~(Q2)] [a~(#2)j ~(~41 ~#2+
#4 a~ ) [c~s(Q2)]~ [ ~ ( # : ) j (~) Q--7
(1)
where #~T ( Q2) is the leading twist term at NLO, #T(I,)(Q2) is the IR-renormalon contribution (fixed by the analysis of the longitudinal channel), while the logarithmic pQCD evolution of the twist-4 (twist-6) multiparton correlation contribution is accounted for by the term [a,(Q2)/a~(#2)] "~') ([as(Q2)/c~(#:)] "~6)) with an effective anomalous dimension 7~4) (3~6)) and the parameter a 4 (a~) represents the overall strength of the twist-4 (twist-6) term at the renormalization scale Q2 =/~2 (for the latter the value # = 1 GeV/c has been adopted). Some of the results based oil Eq. (1) are shown in Fig. 1. Our main results [5] can be summarised as follows: i) the higher-twist contributions to the second moment MT(Q 2) turn out to be very small in the whole Q2-range of the analysis; ii) for n >_ 4 the twist-4 and twist-6 contributions have opposite signs (see Ref. [5]), so that the total higher-twist contribution becomes smaller than its individual terms; in particular, at large Q2 the sum of the twist-4 and twist-6 contributions is just a small fraction of the twist-2 term (G 10% for Q2 ~ n (GcV/c)2); iii) for n >_ 4, the
G. Ricco, S. Simula /Nuclear Physics A666&667 (2000) 165c-168c
167c
IR-renormalon contribution increases significantly around Q2 ~ 1 (GeV/c) 2 becoming of the same order of magnitude of the twist-2 term at N L O in case of higher order moments; the effects from multiparton correlations appear to exceed the IR-renormalon term only for Q2 > 2 (GeV/c) 2 (at n _> 4). Finally, it has been checked that our determination of the multiparton correlation effects is only marginally affected by the specific value adopted for o~(M}) (see Ref. [5]).
. . . . . . . .
0.016
E
0.20
i
.
.
.
.
.
.
.
.
i
0.012
"o
0.15
0.008 0.10 0.004
0.05
0.000
0.00
10
10
Q2 (GeV/c)2
Q2 (GeV/c)2
Figure 1. Twist analysis of the proton transverse moments MT(Q 2) for n = 2 (a) and n = 6 (b). The solid lines are the results of Eq. (1) fitted by the least-x 2 procedure to the pseudo-data (open squares). The dashed lines are the twist-2 contribution, while the dotted and dot-dashed lines correspond to the IR-renormalon and multiparton correlation contributions, respectively.
An interesting feature of our analysis of the transverse moments is that the leadingtwist contribution is extracted from the data and not fixed by calculations based on a particular set of parton distributions. The comparison of our extracted twist-2 term with the predictions based on the GI~V parton distributions [8] is shown in Fig. 2 for Q2 >~ 5 (GeV/c) 2 and n = 6. It can clearly be seen that our results and the GI{V predictions agree quite well in case of the proton, whereas they differ significantly in case of the deuteron. The inclusion of a new empirical determination of the nuclear effects in the deuteron (see Ref. [3] for more details), increases only a little bit the disagreement (see dashed lines in Fig. 2). A possible solution, advocated in Ref. [3], is an enhancement of the d-quark parton distribution at large x. The resulting modified G R V predictions, including also the empirical nuclear corrections in case of the deuteron, are shown by the solid lines in Fig. 2 and agree quite well with our results. Therefore, our extracted twist-2 moments are clearly compatible with the hypothesis of an enhancement of the d-quark distribution at large x. 3. C O N C L U S I O N S The power corrections to the Q2 behaviour of the low-order moments of both the longitudinal and transverse structure functions of proton and deuteron have been analysed using available phenomenological fits of existing data in the Q2 range between 1 and 20 (Ge~Jc) 2. The effects of both target-dependent higher-twists (i.e., multiparton cor relations) and target-independent power corrections (i.e., renormalon ambiguities) have
G. Ricco, ,7. Simula /Nuclear Physics A666&667 (2000) 165c-168c
168c
O.OOS
0.004
. . . .
,
. . . .
,
(a)
. . . .
(b) 0.003
0.004
0.002
0.003 10
15
Q2 (GeV/c)2
5
10
15
Q2 (GeV/c)2
Figure 2. The leading-twist moment #T(Q2) versus Q2 for n = 6. Open dots: twist-2 extracted fi'om our analysis of the proton (a) and deuteron (b) transverse pseudo-data; the errors represent the uncertainty of the fitting procedure corresponding to one-unit increment of the x 2 / N variable. Dot-dashed lines: GRV prediction at NLO [8]. Dashed lines: the same as the dot-dashed lines, but including the empirical nuclear correction as in Ref. [3] in case of the deuteron. Solid lines: the same as the dashed lines, but including the enhancement of the d-quark distribution at large x of Ref. [3].
been determined from the data. The renormalon contribution is able to reproduce the longitudinal channel and its determination is in nice agreement with the results of a recent N L O analysis [4] of neutrino data on x F ~ ( x , (22). The contributions of multiparton correlations to both the twist-4 and twist-6 terms have been phenomenologically determined in the transverse channel and found to have non-negligible strengths with opposite signs. An interesting outcome of our analysis is that the extracted twist-2 term appears to be compatible with an enhancement of the d-quark parton distribution at large x. REFERENCES 1. M. Virchaux and A. Milsztajn: Phys. Lett. B274 (1992) 221. X. Ji and P. Unrau: Phys. Rev. D52 (1995) 72. 2. M. Dasgupta and B.R. Webber: Phys. Lett. B382 (1996) 273. E. Stein et al.: Phys. Lett. B376 (1996) 177. M. Maul et al.: Phys. Lett. B401 (1997) 100. 3. U.K. Yang and A. Bodek: Phys. Rev. Lett. 82 (1999) 2467. 4. A.V. Sidorov: these proceedings and references therein quoted. 5. G. Ricco, S. Simula and M. Battaglieri: Nucl. Phys. B555 (1999) 306. 6. G. Ricco et al.: Phys. Rev. C57 (1998) 356; Few-Body Syst. Suppl. 10 (1999) 423. 7. O. Nachtmann: Nucl. Phys. B63 (1973) 237. 8. M. Gliick, E. Reya and A. Vogt: Z. Phys. C67 (1995) 433.