Quark and gluon distributions and alpha-s from nucleon structure functions at low X

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Physxcs Letters B 309 (1993) 222-230

North-Holland

PHYSICS LETTERS B

Quark and gluon distributions and

from nucleon structure functions at low x

New M u o n Collaboration

Blelefeld Umverslty, CERN, Frelburg Umversxty, Max Planck Instltut fur Kernphyslk, Heidelberg, Heidelberg Umverslty, Malnz Umvers~ty, Mons University, Neuchatel Umverslty, NIKHEF-K, Saclay DAPNIA/SPP, University of Callforma, Santa Cruz, Paul Scherrer Institute,

Tormo University and INFN Tormo, Uppsala University, Soltan Institute for Nuclear Studies, Warsaw, Warsaw University

M Arneodo m,1, A Arvtdson n, B Badelek v, M Balllntun 1, G Baurn a, J Beaufays '2,

I G Bird 1,3, P Bjorkholm n, M Botje e,4, C Brogglm h,5, W Bruckner a, A Brull c, W J Burger e,6, J Clborowskl P, R van Dantzlgl, A Dyrlng n, H Engehen c, M I Ferrero m L Flurl h,

U Gaul d, T GramerJ, D von Harrach d,7, M van der H e u d e n 1,4, C Heusch k, Q Ingram e, K Janson-Prytz n,8, M de Jong f, E M Kabul3 d,7, R Kaiser c, T J Ketel ~, F Klein f,

S Kullander n, U L a n d g r a f c, T Llndqvlst n, G K Mallot f,b, C Marlottl m,9,

G van Mlddelkoop 1, A MllsztajnJ, Y Mlzuno d,l°, J Nassalskl °, D N o w o t n y d,ll, J Oberskl 1, A Pal6 h, C Peronl m, B Povh d,e, R Rleger f,12, K Rlth d,13, K Rohrtch f,14, E R o n d l o °, L Ropelewsklp,3, A Sandacz °, D Sanders 15, C Scholz d, R Schumacher ~,16, R Seltz f, F Sever 1,17, T - A Shlbata e, M Sleblera, A Slmond, A Stamno m, M Szleper °,

Y T z a m o u r a m s dAS, M VlrchauxJ, J L Vudleumler h, T Walcher f, R Wmdmoldersg, A WltzmannC a n d F Zetsche ~

a Btelefeld Umverslty, Physics Department, 4800 Btelefeld, German) 18 b CERN, 1211 Geneve 23, Switzerland

c Frelburg Umverslty, Physics Department, 7800 Fretburg, Germany

18

d M a x Planck Instttut fur Kernphystk, 6900 HeMelberg, Germam' 18 e Heidelberg Umverstty, 6900 HeMelberg, Germany 18

f M a m z Umverslty, Instttutfur Kernphyslk, 6500 Mamz, Germany

18

g Faculte des Sciences, Untversttb de Mons, 7000 Mons, Belgium h Umverstte de Neuchatel, 2000 Neuchatel, Switzerland

N1KHEF-K, P 0 Box 4395, 1009 DB Amsterdam, The Netherlands

19

J DAPNIA, Service de Physique des Parttcules, CE Sac[ay, 91191 Glfisur-Yvette, France k Umverstty o f Cahforma, Institute for Particle Physics, Santa Cruz, CA 95064, USA e Paul Scherrer Instztute, 5234 Vllhgen, Switzerland

m Untverstta dl Tortno, Istttuto dl Flslca, 10125 Tormo, Italy

n University o f Uppsala, Department of Radlatton Science, 75121 Uppsala, Sweden o Soltan Institute for Nuclear Studzes, 00681 Warsaw, Poland 2°

P University of Warsaw, 00681 Warsaw, Poland 2°

Received 13 April 1993 Editor L Montanet

The Q2 dependence of the structure functions F~ and F2 d recently measured by the NMC is compared with the

predictions of perturbatlve QCD at next-to-leading order Good agreement is observed, leading to accurate determi-

nations of the quark and gluon distributions in the range 0 008 ~< x ~< 0 5 The strong coupling constant is measured

from the low x data, the result agrees with previous determinations

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Volume 309, number 1,2 PHYSICS LETTERS B 8 July 1993 1. Introduction

In a previous letter [1] the New M u o n Collabo- ration (NMC, C E R N - N A 3 7 ) p r e s e n t e d proton (F~) a n d deuteron ( F ( ) structure functions obtained from simultaneous measurements of deep inelastic m u o n scattering on hydrogen a n d deuterium targets at in- cident m u o n energies of 90 a n d 280 GeV The data cover a wide range m the Bjorken scaling variable x a n d m the square of the f o u r - m o m e n t u m transfer _Q2 ( 0 0 0 8 ~< x ~< 0 5 a n d 0 8 ~< Q2 ~ 4 8 G e V 2) An i m p o r t a n t feature of these data is their extension w~th good accuracy to low values of x In this letter we present the results of an analysis of the N M C data in terms of q u a n t u m chromodynamlcs ( Q C D )

I n the Q C D parton model the x a n d Q2 dependences of F2 are related to those of the quark a n d gluon distributions The

Q2

dependences of these &strlbu- tlons, due to the processes of gluon radiation a n d q~

pair creation, are obtained from the Q C D evolution

1 Now at Dlpartlmento dl Flslca, Unlverslta della Cal- abria, 1-87036 Arcavacata dl Rende (Cosenza), Italy 2 Now at Trasys, Brussels, Belgmm

3 Now at CERN, 1211 Geneve 23, Switzerland

4 Now at NIKHEF-H, 1009 DB Amsterdam, The Netherlands

5 Now at INFN, Laboraton Nazionah del Gran Sasso, 67010 Assergl, Italy

6 Now at Unlvers~te de Geneve, 1211 Geneve 4, Switzerland

7 Now at University of Mamz, 6500 Malnz, Germany 8 Now at DESY, 2000 Hamburg 52, Germany

9 Now at INFN - Istltuto Superlore dl Samta, 1-00161 Roma, Italy

10 Now at Osaka University, 567 Osaka, Japan 11 Now at SAP AG, 6909 Walldorf, Germany

12 Now at Ploenzke Informatlk, 6800 Mannhelm, Germany

13 Now at University of Erlangen, 8520 Erlangen, Germany

14 Now at IKP2-KFA, 5170 Juhch, Germany

15 Now at University of Houston, TX 77204-5504, USA, funded by NSF and DOE

16 Now at Carnegie Mellon Umverslty, Pxttsburgh, PA 15213, USA

17 Now at ESRF, 38043 Grenoble, France

18 Supported by Bundesmlnlsterlum fur Forschung und Technologie

19 Supported in part by FOM, Vnje UmverslteIt Amster- dam and NWO

20 Supported by KBN grant no 2 0958 9101

equations [2], using their x dependences at a given Q2 as inputs Perturbatlve Q C D does not predict the x dependences In leading order (LO) the Q2 evolution of a linear c o m b i n a t i o n F of quark (q) a n d an- tiquark (~) distributions is given by

1

OlnQ 2 - 2~ pqq y F(y,

~2

f l

+ Z ( G + - d ~ ) / T P q g ( ~ ) G ( y , Q2)],

(1)

l = l

with

f

F(x,Q 2) = Z {c,q,(x, Q2)

+ F , ~ , ( x , Q2)} (2)

t = l

Here the sum runs over the active flavours t = u, d, s, , as is the strong coupling constant,

Pqq

and

Pqg

are Q C D sphttlng functions a n d G is the gluon distribution A similar evolution equation exists for G

The function F can be expressed as a linear combi- nation of a flavour non-slnglet distribution, for which

(c, +

F,) = 0, and the slnglet distribution which as the sum of all quark a n d antlquark distributions (c, = F,), see e g ref [3] Whereas the Q2 evolution of a non-slnglet distribution does not depend on G (the second term on the right h a n d side o f e q (1) vanishes), the Q2 evolution of the smglet distribution is coupled to that of the gluon

In a d & t l o n to the logarithmic Q2 evolution predicted by perturbatlve Q C D (eq (1)), non- logarithmic contributions to the Q2 dependence of F2 come from target mass effects a n d the interaction of the struck quark w~th the spectator quarks (higher twist effects) At a given x, they both behave like power series in 1/Q2 a n d may thus become ~mportant at low values of Q2

2. QCD analysis

The essence of a Q C D analysis of structure functions is the comparison of their Q2 dependence with the prediction of perturbatlve Q C D F r o m parametrlsatlons of the effectave smglet quark, non-slnglet

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Volume 309, number 1,2 PHYSICS LETTERS B 8 July 1993 quark a n d gluon distributions at a given scale Q2

- qSi(x, Q2), qNS(x, Q2)

a n d

G ( x , Q ~ ) -

a n d from the value of the strong coupling constant C~s(Q2), one can compute structure functions at any x a n d Q2 using the Q C D evolution equations In a Q C D fit to a set of structure function measurements the coeffi- cients of these parametrlsatlons a n d C~s are adjusted to give the best agreement between the measured a n d the computed structure functions over the whole x and Q2 d o m a i n In such a procedure, the value of Q2 is arbitrary, in practice one often chooses a value typical of the data

To perform this task, we have used a computer program developed from that of ref [4] This program performs a vectorlzed fully numerical integra- tion of the evolution equations in next-to-leading order (NLO) in the MS renormallsation a n d factorisa- tion schemes [5] In the program, the Q2 evolution ( " r u n n i n g " ) of the strong coupling constant was calculated from the NLO r e n o r m a h s a t i o n group equation, with the flavour thresholds treated as described in ref

[6] To calculate the Q2 evolution of the charmed sea quark distribution, which differs from that of quasi- massless quarks (u, d and s), the prescription of ref

[ 7 ] was adopted

The fit was performed simultaneously on the measured values of F2 p, which has both flavour slnglet and non-singlet components, a n d of F2 d = (F~ + F2 n)/2, which is nearly a pure slnglet structure function This allows a reliable d e t e r m i n a t i o n of both the slnglet a n d non-slnglet quark distributions to be made We have checked that the small non-slnglet c o m p o n e n t of F d, proportional to qs - qc, changes the Q2 evolution of F2 negligibly Thus, it proved more practical to deter- mine the deuteron quark distribution

qd (X, Q2), in-

stead of the slnglet distribution qSi (x, Q2 ), from the Q C D fits We asssumed that the gluon distribution is the same in the proton a n d the deuteron, this is com- patible with recent experimental results [8]

The value of Q~ was chosen to be 7 GeV 2 a n d the parametrlsatlons used in the fit are

x q N S ( x , Q ~ ) = Ax'~(1 - x ) '8,

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x q d ( x , Q 2) = B x r ( 1

- x ) a ( 1 + blv + b2v2),

w i t h y = 0 l - x , (4)

x G ( x , Q 2)

= C(1 - x ) ~ ( 1 +

ClW + c2w 2 + £'3'//33),

with w = 0 1 ln(1 + e l°-l°°x) (5) The variable w in eq (5) was chosen such that It differs from zero only for x < x0 = 0 1, so that the behaviour of the gluon distribution at x > x0 has the usual form proportional to (1 - x ) " It was ver- ified that these parametrxsations are flexible enough to describe Q2 values other than 7 GeV 2, adding ex- tra parameters or using other f u n c n o n a l forms does not significantly improve the quality of the fit In the fit the m o m e n t u m sum rule was imposed, that ~s the slnglet quark distribution a n d the gluon distribution were c o n t i n u e d to x = 0 a n d 1, a n d their integrals required to add up to unity f0' x(qSl + G) dx = 1 The sensitivity of the results to this assumption is discussed later

The effect of higher twist contributions on the Q2 dependence of F2 cannot be calculated from theory It was taken into account in the following way The functions fitted to the data were parametrised as

F 2 ( x , Q 2) = F2LT(x,Q 2)

{1 +

H ( x ) / Q 2} ,

(6) where F LT obeys the NLO Q C D evolution equations a n d

H ( x ) / Q 2

is a phenomenologlcal description of the twist-four contribution #~ The limited range in Q2 does not allow H (x) to be unambiguously determined from the present data It was therefore kept f x e d in the fit A b o v e x = 0 2,

H ( x )

was taken from ref [9], averaged over the proton a n d the deuteron At lower values of x,

H ( x )

was linearly extrapolated to x = 0 The value o f H ( x = 0) --- - 0 13GeV 2 gives the best agreement between the data a n d the result of the Q C D fit With this choice, higher twist contributions are moderate or small in the entire kinematic range of the data The sensinvlty to alternative extrapolations of H (x) at small x IS discussed below

Target mass corrections [ 10 ] were calculated from the measured structure functions a n d taken into account, they are small in the kinematic range of the data Corrections for Fermi m o t i o n in the deuteron were estimated to be small a n d were not applied Also shadowing in the deuteron was not taken into account

#1 The function

H ( x )

may also partly describe next-to- next-to-leading order QCD contnbunons or saturation effects m parton densmes

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Volume 309, number 1,2 PHYSICS LETTERS B 8 July 1993 We have excluded from the fit data with Q2 < 1

GeV 2 No further cuts on the data were made I n ref [1], the m e a s u r e m e n t of F~ a n d F d was oh- t a m e d from the cross sections using a p h e n o m e n o - logical p a r a m e t n s a t i o n [ 11 ] for R, the ratio of lon- gitudinally to transversely p o l a n s e d virtual photon absorption cross sections We have checked that us- mg the Q C D prediction for R [12] instead does not significantly affect the results of our analysis

Values of the fitted parameters in eqs ( 3 ) - ( 5 ) were obtained from a Z 2 m l n l m l s a t l o n procedure with the weights computed from statistical errors only The relative n o r m a h s a t l o n J N between the 90 a n d 280 GeV data was a free parameter in the fits a n d the quantity (tSN/AN) 2 was added to the Z 2, where AN = 2 1%

is the estimated uncertainty in this relative n o r m a h - satlon I1 ] The treatment of other systematic errors is discussed in the next section

Table 1

The values of the parameters of the fitted distributions xq Ns, xq d and xG, eqs (3)-(5) Figures under "Central value"

correspond to the result of the fit with as fixed to 0 240 at 7 GeV 2 The columns "Lower hmlt" ("Upper hmlt") correspond to a parametnsatlon of the lower (upper) bound of the error bands on the dlstnbuUons shown m fig 2 Note that there are twelve independent parameters because the smglet quark and gluon distributions are required to satisfy the momentum sum rule

Lower limit Central value Upper hmlt

A 2 249 2 774 3 700

c~ 0 892 1 003 1 148

fl 2 413 2 787 3 271

B 1 900 2 036 2 390

y - 0 059 - 0 037 0 022

J 3 725 4 159 4 675

b~ - 2 831 - 3 086 - 2 986

b 2 1 978 4 664 8 640

3. Results on the quark and gluon distributions We performed the Q C D analysis m two parts In the first one, described in this section, the value of as was fixed to obtain the quark a n d gluon d~strlbu- tions with the best precision We used the value of a s ( Q 2 = 7 GeV 2) = 0 240, which corresponds to as ( M 2 ) = 0 113, the average of measurements from deep inelastic scattering [ 13 ], and which agrees with the r e c o m m e n d e d value of ref [ 14]

In fig 1 the data are presented together with the result of the fit The sohd curves correspond to the Q C D fit, including contributions from higher twist terms (see eq (6)) This fit provides a good overall description of the data (zE/dof = 333/239, statistical errors) The fitted relative n o r m a h s a t l o n of the 90 a n d 280 GeV data sets was found to be 1 018, in the fit (and also m fig 1 ) the 90 GeV data were lowered by 0 7%, a n d the 280 GeV data raised by 1 1% This is within the n o r m a h s a t i o n errors given in [l ] To illustrate the importance of higher twist effects, F LT as defined in eq (6) is also shown m fig 1 (dashed curves) The fitted values of the parameters of x q d (x, Q2), xqNS (X, Qo 2) a n d x G (x, Qo 2) in eqs ( 3 ) - ( 5 ) are given in table 1

F o u r kinds of error contribute to the uncertainty in the results of the fit

C 3 733 3 781 5 332

r/ 6 391 7 427 10 610

cl 0 205 - 0 105 - 1 515

c 2 0 068 0 385 2 679

c 3 - 0 029 - 0 283 - 1 669

(i) The statistical error corresponding to a Z 2 in- crease o f z 2 / d o f

( n ) The experimental systematic error obtained by repeating the fit with F2 offset according to each source of systematic error [1] in turn a n d adding the resulting deviations in quadrature This procedure takes into account correctly the correlations for each source of systematic error, as described in the preprlnt version of ref [ 1 ] This error includes the effect of a +2% overall normallsation uncertainty

(ni) An error due to uncertainty in the continua- tion of the distributions into the u n m e a s u r e d region, x = 0 - 0 008, which contains about 5% of the nucleon m o m e n t u m A 100% error was assigned to this estimate The resulting uncertainty of the fits was deter- m i n e d by repeating them with the m o m e n t u m sum constrained to 1 05 and 0 95

(iv) Errors due to the uncertainties in C~s and higher twist effects These are discussed below

We quote as the error on the parametrisatlons of the quark and gluon distributions the quadratic sum of contributions (1)- (iu) only

The non-singlet quark distribution x q Ns resulting

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Volume 309, number 1,2 PHYSICS LETTERS B 8 July 1993

C) L ~

0 × L ^TM 0 5

O~

N M C ~ = o oos

p r o t o n / ( X 4 0 "~

x = O 0 "

( X l O )

I i i i , i i i ,

1 10

Q 2 k O e \ 2)

x = 009 [ x 7 5 )

~ = 0 1 1 ( X 5 2 )

, ~ • t ,~ x = 0 1 4

( X 3 7 )

t - - e * , it---ira. , • ~ x = 0 1 8

( X 2 5 )

x = 0 225

~ x 1 7 )

x = 0 275 k X 1 2 )

N M C p r o t o n

x = 035 k X l O )

x = o 5 o ( x l o )

, , , I , , , , , ,i,I

1 10

i , , i i

100 02 ( O e V ~')

0 . L

g 5

N M C ~ = o o o 8

, , ( x 4 o )

* ( x ~ 2 )

× = 0 0 7 k X l O )

i i i i , i

1 10

O 2 ( C e V q

C~

2

0 1

x = 009 ( X 7 5 ) ,11 x = O l l

( X 5 2 )

x = 0 1 4 ( X 5 7 )

× = 0 1 8

£ × 2 5 )

x = 0 225 ( × 1 7 )

x = 0 275 { X 1 2 )

~ = 0 3 5 ( X l O )

x = 0 5 0

( X l O ~

, , i I i i J t iii I , i i i . . . . ,

1 10 100

O 2

(CeV~

Fig 1 The structure functions F~ and F2a measured at mmdent muon energms of 90 and 280 GeV The 90 (280) GeV data are renormahsed by 0 993 (1 011) The errors shown are statlsUcal only The sohd curves correspond to the result of the Q C D fit described m sectmn 3 The dashed curves are the result of the same fit with the higher twist contnbutmns subtracted

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Volume 309, number 1,2 PHYSICS LETTERS B 8 July 1993

0 5

0 4

0 3

0 2

0 1

6 (_9

x

Q2=7 OeV 2

(a)

i i i i , i i , i , I i i , i i , , ,

0 0 1 0 1

X

8 (c)

0

o 0 1 0 1 X

Q~ = ~

0 0 1 0 1

2 5

~ET X

1 5

0 5

(b) Q2=7 OeV 2

0 0 1 0 1

X

Fig 2 (a) The non-smglet quark distribution xq Ns, (b) the deuteron quark dlstrlbunon

xqd

and (c) the gluon distribution

xG,

all at a fixed value of Q2 = 7 GeV 2, obtained from this QCD analysis The central curves correspond to the result of the fit described in section 3, the error is indi- cated by the bands around the curves In the insert the gluon dlstnbunon from this analysis, evolved to Q2 = 1 25 GeV 2 and 48 GeV 2, is shown The wiggle Is an artefact of the parametrxsatlon used

from the fit is shown m fig 2a at Q2 = 7 GeV 2 It corresponds to 3 (F~ - F2 ~ ) a n d can be used to estimate the Gottfried sum,

f2 qNS(x)/3

dx = 0 2 4 3 ± 0 030, in agreement with the result of ref [ 15 ] As F~ - F2 n Is a non-slnglet structure function, its Q2 evolution is determined by the strong couphng constant only We have checked that the x dependence of the slopes d(F2 p - F2 ~ ) / d l n Q2 is consistent with the Q C D prediction The present data on F~' - F2" are not accurate enough however to significantly constrain as

The x dependence of the fitted deuteron quark dis- t n b u t l o n

xq d (x, Q~ )

(corresponding to

~ F, d )

₅ ₂ a n d of the gluon distribution

x G (x, Q2)

are shown in figs 2b

and 2c The deuteron quark (gluon) distribution IS extracted with an uncertainty of about i 5 % (+20%) at x = 0 01 At higher x the uncertainty is smaller The m o m e n t u m fracuon carried by the quarks was found to be 0 55:1:0 02 at Q2 = 7 GeV 2 At Qz =

1 25 a n d 48 GeV 2, the extreme values of Q2 m our data, this f r a c n o n evolves to 0 60 a n d 0 52, respec- tively The fitted gluon distribution, evolved to 1 25 a n d 48 GeV 2 is shown in the insert of fig 2c, at the lowest x it changes by an order of magnitude

The sensitivity of the gluon distribution to the assumed value of

as

is displayed in fig 3a The dashed curves are the results of fits with c~s (7 GeV 2 ) fixed to

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Volume 309, n u m b e r 1,2 PHYSICS LETTERS B 8 July 1993

C.D

x

(a)

Q2=7 OeV 2 C~s(7 CeV2)=O 215

~ a s ( 7 GeV2)=O 240

~xs(7 CeV2)=O 265

0 0 1 0 1

X

x (b)

5 Q2=7 OeV2

no higher twist

4

3

2

T

0

0 0 1 0 1

X

Fig 3 (a) The sensitivity of the gluon dlstnbutlon to the assumed value of as The solid curve is the result of the QCD fit with as fixed to 0 240 at 7 GeV 2 The error band is ~dentlcal to that of fig 2c The two dashed curves are the results of fits with C~s fixed to 0 265 and 0 215 (b) The sensitivity of the gluon d i s t n b u n o n to higher twist effects The dashed curve corresponds to the gluon distribution obtained from a fit with all higher twist terms fixed to zero

(H(x)

~ 0) and as = 0 240 The solid curve and the error b a n d are as in (a)

0 215 a n d 0 265 w h i c h reflect t h e t o t a l e r r o r o n c~s f r o m d e e p i n e l a s t i c s c a t t e r i n g e x p e r i m e n t s [ 13] T h e u n c e r t a i n t y i n C~s h a s a n e g l i g i b l e e f f e c t o n t h e q u a r k d i s t r i b u t i o n T h e s e n s i t i v i t y o f t h e g l u o n d i s t r i b u t i o n

-4

NMC

/q SLAC+BCDMS

Q2=20 OeV 2

CDHSW

/

001 01

X

Fig 4 The gluon distribution from this analysis compared to two previous determinatxons in deep inelastic scattering, from the BCDMS and SLAC hydrogen and deuterium data in NLO [9] and from CDHSW iron data in LO [16]

to h i g h e r t w i s t effects is s h o w n i n fig 3 b T h e d a s h e d c u r v e 1s t h e r e s u l t o f a fit w i t h n o h i g h e r t w i s t t e r m s T h e effect o f s u c h t e r m s o n t h e q u a r k d i s t r i b u t i o n is m u c h s m a l l e r T h e s e n s i t x v l t l e s s h o w n i n figs 3a a n d 3 b r e p r e s e n t u p p e r l i m i t s a n d are c o m p a r a b l e to t h e t o t a l e x p e r i m e n t a l e r r o r

T h e g l u o n d i s t r i b u t i o n f r o m t h i s a n a l y s i s is c o m - p a r e d i n fig 4 t o p r e v i o u s d e t e r m i n a t i o n s f r o m d e e p i n e l a s t i c s c a t t e r i n g d a t a , f r o m B C D M S a n d S L A C m N L O [9] a n d f r o m C D H S W m L O [ 1 6 ] T h e i m - p r o v e m e n t i n p r e c i s i o n at low x is a p p a r e n t

W e o b s e r v e t h a t t h e g l u o n d i s t r i b u t i o n o b t a i n e d i n a s i m i l a r k i n e m a t i c r a n g e b y N M C f r o m a n a n a l y s i s o f i n e l a s t i c

J~ ~'

p r o d u c t i o n [ 17 ] a g r e e s w i t h t h e p r e s e n t r e s u l t D e t e r m i n a t i o n s o f t h e g l u o n d i s t r i b u t i o n f r o m t h e o b s e r v a t i o n o f d i r e c t p h o t o n s i n h a d r o n - h a d r o n i n t e r a c t i o n s [ 1 8 ] , o f t e n o b t a i n e d a t l a r g e r x o r Q2, are also i n a g r e e m e n t w i t h t h e p r e s e n t r e s u l t

4. Measurement of the strong couphng constant and test of QCD

I n t h e s e c o n d p a r t o f t h e a n a l y s i s , we d e t e r m i n e d

O~s(Q 2)

f r o m t h e N M C d a t a , b y l e a v i n g it as a free p a r a m e t e r i n t h e fit T h i s r e s u l t e d i n a v a l u e f o r t h e

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Volume 309, number 1,2 PHYSICS LETTERS B 8 July 1993 strong couphng constant

as(7 GeV 2) = 0 264 ± 0 018(stat ) + 0 070(syst )

+ 0 013(h t ) (7)

The systematic error includes sources of uncertainty listed u n d e r ( n ) , ( n i ) in the previous section The d o m i n a n t sources of systematic error are the uncertainties in the spectrometer acceptance correction a n d on the energy calibration [ 1 ] The "h t " error results from the uncertainty in the higher twist terms, it was obtained from fits where the function H ( x ) of eq (6) was changed at x > 0 20 within the errors given in

[9], and at lower x such that

H ( x

= 0) varied between 0 a n d - 0 25 GeV 2

The present result corresponds to a s ( M z 2 ) = 0 117+0011 It is consistent with other measure- * ' - - 0 0 1 6

ments of as [19], in particular with the present average from deep inelastic scattering,

a s ( M 2) =

0 113 + 0 002 ( e x p ) [ 13], used m the previous section for the d e t e r m i n a t i o n of quark a n d gluon dis- t n b u t l o n s We have checked that applying different Q2 cuts on the data does not significantly change the value of as obtained Uncertainties of theoretical origin in as are d o m i n a t e d by the arbitrariness of the choice of the r e n o r m a h s a t i o n a n d factonsatlon scales, these have been studied in refs [9,20] a n d are small compared to our experimental error

An imperfect representation of the x dependence of F2 in the fit may bias the result on as To check for such a bias, the fit was repeated with the gluon a n d quark distributions allowed to adjust to an optimal value in each b i n of x separately No statistically significant adjustments were found n o r was the Z 2 substantially improved

The agreement over the full x range of the Q2 dependences of the data with those predicted by Q C D is the i m p o r t a n t illustration of their consistency For that purpose, the average logarithmic slopes

d l n F 2 / d l n Q 2

were determined in each b i n of x separately, both from the data a n d the Q C D fit #2 These are shown in fig 5, the points correspond to the

Q2

evolution of the data a n d the solid curves to

#2 In the latter case, an error equal to that of the measured F2 was asmgned to the predicted values, in each bin of x and

Q2

~o2 2

O1

0

(31

..~ F2_ F2LT( 1 +H(x)/Q2) N M C (a)

p r o t o n

It) r2L'

1

0 1 0 2 0 3 0 4 0 5

x

L 2 (b)

7z~o3

t/~[~ r2=r2L'(l+H(x)/Q) NMC

d e u t e r o n o. 02 [ ~

c

o

_Ol :

, , Or2 , , , I , , , , I i , , L

0 1 ' ' 0 3 0 4 0 5

x Fig 5 The logarithmic slopes d In

1;'2/d

In Q2 of the data

(circles) compared to those of the QCD fit of secUon 4 (solid curves) for (a) the proton and (b) the deuteron The errors shown are statistical only The dashed curves correspond to the QCD predlcuon without higher twist terms (see text) The dotted curves correspond to the Q2 evolution from quarks only

that of the fit Good agreement IS observed In fig 5, the dashed curves correspond to the Q2 evolution of

F LT,

see eq (6) The difference between the dashed a n d solid curves indicates that the higher twist con- t n b u t l o n to the logarithmic slopes is moderate

The dotted curves in fig 5 Indicate the Q2 evolution due to quarks only and the shaded areas between

(9)

Volume 309, number 1,2 PHYSICS LETTERS B 8 July 1993 the dashed a n d dotted curves represent the contribu-

tion of gluons It is clear that for most of the N M C data, the Q2 evolution is driven by the gluon distribution thus the present analysis is sensitive to the prod- uct c~s ×

G(x)

As the gluon distribution can be constrained by the m o m e n t u m sum rule, C~s can be determined In previous Q C D analyses of deep inelastic scattering data, as was mainly constrained by the

Q2

evolution at high x, where the gluon dtstrlbutlon has a small influence The present analysis extends the de- t e r m i n a t i o n of c~s a n d the test of Q C D to the low x d o m a i n

5. Conclusions

We have presented a next-to-leading order Q C D analysis of the F2 structure functions of the proton and the deuteron recently obtained by the N M C The

Q2

evolution of F2 p a n d F2 d is in good agreement with perturbatlve Q C D down to Q2 = 1 GeV 2, with only a moderate c o n t r i b u t i o n from higher twist terms The evaluation of the strong couphng constant as at low x a n d Q2 agrees with previous determinations in deep inelastic scattering We have obtained an accurate m e a s u r e m e n t of the quark and gluon distributions down to x = 0 008

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