Thermodynamical analysis of the EMC effect

Volume 154B, n u m b e r 4

PHYSICS LETTERS

2 May 1985

THERMODYNAMICAL ANALYSIS OF THE EMC EFFECT C. A N G E L I N I and R. PAZZI Dipartimento di Fisica dell'Universiti~ di Pisa, Pisa, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy Received 11 December 1984

Assuming that the sea quark distribution vanishes for x > 0.3, we analyse the functions measured by the European Muon Collaboration in the framework of a quarks. The experimental ratio F2Fe(x)/F2°(x) is well reproduced over the whole x distributions at different temperatures T and confinement volumes V. We obtain T o

Recently, the European Muon Collaboration (EMC) [ 1] has pointed out that the nucleon structure function F2(x, Q2) shows a different x dependence when measured i n / l - F e or in/a-D deep inelastic scattering. The ratio FFe/F~2 seems to be a linear function o f x and shows a very weak Q2 dependence. A general review of this effect as compared to the other experiments performed during the last year is contained in ref. [2]. Several models [3] ,x have been proposed to explain this effect. In particular, as a consequence of dynamical effects, it has been suggested that the quark confinement size is larger in heavy nuclei than in free nucleons. In this letter we support this hypothesis with an analysis of the structure functions using a thermodynamical model, that we already applied successfully to the valence quark distributions

[5,61. It is generally believed that the sea quark distribution vanishes for x > 0.3. Therefore, in this region, F2(x ' Q2) would behave like a valence quark distribution and our model could be safely applied. Since the rather limited number of data points forx > 0.3 is not sufficient to constrain the fits at fixed Q2, we analyse the data in x (x > 0.3) and Q2 variables using the same QCD approximate evolution we applied in ref. [6]. Hence we fit the F Fe 2 (x, Q 2 ) and F~2(x, Q2) ,1 For a general review see, for example, ref. [4].

328

FzVe(x, Q2) and FzD(x, Q2) structure thermodynamical model of the valence range by the ratio of two valence quark - T v e - 3 MeV and V v ¢ / V D --1.3.

structure functions with the following formulae: F 2 ( x > 0.3,02) -- xqv(x, Q2), qv(X ' Q2) ~_qv(X ' Q2)

+

s

1

dy

2p

x

f 7 qv y, Qo) qq(/Y)

,

(1)

x

qv(X ' Q2) = 2rtMCT 2 [e-aX(ax + 1 ) - e-a(a + I)], (2)

with s =ln/-ln'Q2A21/¢ / ~\ \In(Q2/A2)] ' b = g/4rt,

a=M/2T

with 3 flavours,

where T is the quark gas temperature, M the nucleon mass, x the Bjorken variable and C is a normalization constant which is related to the nucleon effective volume V via the relation C = gV/(2rr) 3 , where g is the multiplicity factor* 2. Since the EMC effect is apparent in the ratio of the F2(x , Q2) structure functions measured on different targets, any systematic difference has to be avoided in fitting the two data sets with the same model. For this purpose we use only points at nearly the . 2 In refs. [4,5 ] we used A = 2 n M C T 2 as normalization constant.

0370-2693/85•$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 154B, number 4

PHYSICS LETTERS

2 May 1985

Table 1 The results of the fits of the EMC F2(x, Q2) on iron and deuterium with eqs. (1), (2) in the text for x > 0.3 and 15 < Q2 < 90 (GeV/c) 2 . Only statistical errors are taken into account. Q2

T

C

A2

(GeV/c) 2

(MeV)

(GeV) -3

(GeV/c) 2

x2/dof

CFe/CD

X 104 10

Fe D

49.7±0.6 53.0±0.9

214±10 160±14

47± 50 5± 17

31/18 31/19

1.34±0.13

20

Fe D

48.8±0.4 52.2±0.9

218± 8 163±13

71± 75 7± 24

31/18 31/19

1.34±0.12

50

Fe D

47.9±0.4 51.4±0.8

221± 8 166±13

120±130 11± 38

30/18 31/19

1.33±0.11

90

Fe D

47.4±0.3 51.0±0.8

222± 8 167±13

170±190 14± 52

30/18 31/19

1.33±0.11

same x and Q2, essentially in the x and Q2 region spanned by the deuterium data ,3. Furthermore, as in ref. [6], we use the data in a limited Q2 range [ 15 < Q2 < 90 (GeV/c) 2 ], because of the approximation made in the Altarelli-Parisi [7] equation (1). We take T, C and A 2 as free parameters and a starting value Qo2 = 10 (GeV/c) 2 (A 2 is left as free parameter because it is expected to depend on our approximation). The fits are repeated with different values of Q2 [20, 50, 90 (GeV/c) 2] in order to investigate, in the range considered, the dependence on Q2 o f T and C. The results of the fits are reported in table 1 and the fit at Q2 = 20 (GeV/c) 2 is displayed in fig. 1 as an example. We observe that the temperatures and the normalization constants, though changing with Q2, satisfy the relations: T D - TFe -----3 + 1 MeV and CFe/CD = VFe/F D ~ 1.3 + 0.1 independent o f a 2 and with a reasonable probability. In addition we notice that the values o f A change with Q2 (especially for the Fe data) and have different averages [
.18 36

'.',

........

,

7/,

X = 0.35

.14

.1(1

X = 0.45

.08 .06 .06

½ ! {~

X= 0.55

.04

.0/. .02

X= 0.65

.00

F2Fe (X,O)2

,0

. . . . . . . . . . . . . . . . .

10o'

F;(X,O 2)

','0

........

' 100

O2 (GeV/c)2 Fig. 1. The F~Fe(x, Q2) and FD(x,Q2) structure functions measured by the EMC for x > 0.3. The curves are the QCD fits in the approximation discussed in ref. [6] with Qg = 20 (GeV/ c)2. Only data with 15 < Q2 < 90 (GeV/c) 2 are used. [8], agree with our average values and do not show any dependence on Q2. The results for the ratio xqFe(x, Q2)/xqD(x, Q2) at various Q2 as a function o f x are shown in fig. 2a together with the experimental data. The difference of about 3 MeV in the temperatures is sufficient to 329

i

i

a)

b)

1.2 x

1.1

1,0

0.9

0.8

t

o12 To ---// (MeV)

o., .

i

o'.6

.

.

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PHYSICS LETTERS

Volume 154B, number 4

i 0.4

02

.

.

I 0.6

c)

53

the ratio of the valence parts o f F Fe and FI~2. We verified that this unexpected result is independent of our approximations, by making the following checks: (i) we used A2e = (A~e)and A 2 = (A2 ) constant with Q~: the trend is about the same and depends less on Q~; (ii) since the errors of deuterium data are larger than those of iron, we forced the deuterium data to have the same QCD scaling violations as iron, that is assuming A 2 = A2e = (A2Fe): the result is again qualitatively the same as in case (i) but quite Q2 independent ,4. Thus it appears that, if we neglect the possible effects of our approximations, the ratio of the thermodynamical valence quark distributions in iron and in deuterium is equal to FFe/FI~2 for all x and, according to the experiment, independent of Q2. This result enables us to fit directly the ratio FFe(x)/FD(x) over the whole x range by the formula*S:

R(x) =

rFe

VD T2

e-"rex(aFeX ÷ 1) - e-are(aFe ÷ 1)

e-aDx(aD x + 1) - e-aD(aD + 1) (3)

5;

with 51

ave,O = 11/1/2TFe,D . 50

J"2

'

]

I

do

I

TF, ( M e V )

Fig. 2. (a) The EMC ratio FFe/F 2 2D averaged over Q2. The curves are the model predictions using the fit results reported in table 1. The arrow indicates the direction o f increasing o f Qg. (b) The same data as in (a). The curves are the EMC straight line fit (dotted) and the model predictions (full) fitting the ratio as discussed in the text. (c) The comparison between the temperatures as obtained in the x and Q2 fits (the points) and the ones as obtained fitting the ratio (dashed area).

reproduce the slope of the straight line fit FFe/FI~2 = a + bx, published by the EMC collaboration with b = - 0 . 5 2 +- 0.04(stat.) +- 0.21(syst.). From the ratio

In our model, by such a fit, we would estimate the temperatures TFe , T D and the ratio VFe/V D corresponding to the average experimental Q2 for iron and deuterium, respectively. The fit is found to be not constrained. Indeed we obtain VFe/V D = 1.30 + 0.02 with very good probability (x2/dof = 2.4/6) but the values obtained for the temperatures depend on the starting values used in the fit procedure. If we fix one of the two temperatures to one of the values reported in table 1 at a given Qo2, then the fitted value for the temperature corresponding to the other target is consistent with that reported in the table at the same Qo2. If we vary the fixed starting value of one of the temperatures between its minimum [at Q2 = 90 (GeV/c) 2 ] and maximum [at Q2 = 10 (GeV/c) 2] reported in

CFe/CD we estimate that the confinement volume in iron is about (30 + 10)% larger than that in deuterium. This result does not differ significantly from existing estimations [3,4] deduced as a consequence of dynamical effects on quark distributions. In addition, we observe that even for x < 0.3 the experimental ratio seems to be well reproduced by 330

4:4 As a last check we repeat the fits using points at x > 0.4 only. The temperatures decrease by about 2 MeV, the errors enlarge but we rind again T~ - T F e ~ 3 MeV and VFe/VD ~- 1.3. The ratio F~e/F~ is reproduced by the ratio o f valence distributions again. , s If we use qv = 2~rMCT2e-aX(ax + 1), which is without any kinematical limit, we have the same R(x) numerically.

Volume 154B, number 4

PHYSICS LETTERS

table 1, then the value obtained for the other temperature is always consistent with the relation T D - TFe 3 MeV; the ratio VFe/VD remains constant and equal to 1.30 +- 0.02 with the same X2 . Since in our picture the scaling violation essentially is apparent in an effective temperature dependence on Q2 [6], we can interpret these (infinite) pairs o f temperatures as the various quark gas temperatures one can estimate, for each target, at different, fixed Q2. The ability o f the model to reproduce satisfactorily the experimental ratio (which is averaged over Q2) with an infinity of pairs T F e, T D is consistent with the fact that this ratio is Q Z i n d e p e n d e n t . Fig. 2b shows how eq. (3) reproduces the EMC ratio by the same curve for any possible pair o f temperatures. The agreement is very good, especially for 0.1 < x < 0.4 where our fit is practically identical to the EMC linear fit ( d o t t e d line). Fig. 2c shows the consistency between the pairs TFe, T D obtained in the x and Q2 fits (points) and the possible ones within the errors obtained fitting the ratio (dashed area). In conclusion, we have verified that our thermodynamical model for valence quark distribution can describe the F2(x > 0.3, Q2) structure function measured by the EMC in iron and deuterium target at different temperatures and normalization constants. Furthermore, we have some indication that the ratio FFe/F D 2 2 behaves as a ratio o f valence quark distribu-

2 May 1985

tions over the whole x region. This would imply that the sea quark distribution in iron changes b y the same ratio as the valence quarks. The confinement volume o f quarks in iron is found to be about 30% larger than that in deuterium. We would like to thank R. Del Fabbro and B. Saitta for carefully reading the manuscript and for discussions. In particular we thank B. Saitta for drawing our attention to the opportunity o f using the same Q2 range for iron and deuterium data in our fits.

References [1 ] The EMC CoUab., J.J. Aubert et al., Phys. Lett. 123B (1983) 275. [2] SJ. Wimpenny, Invited talk 10th Intern. Conf. on Particles and nuclei (Heidelberg, 1984) CERN-EP/84-115. [3] R.L. Jaffe, Phys. Rev. Lett. 50 (1983) 228; F .E. Close, R.G. Roberts and G.G. Ross, Phys. Lett. 129B (1983) 346; R.L. Jaffe, F.E. Close, R.G. Roberts and G.G. Ross, Phys. Lett. 134B (1984) 449. [4] O. Nachtmann, Invited talk Neutrino 84 Conference, (Nordkirchen, 1984). [5] C. Angelini and R. Pazzi, Phys. Lett. l13B (1982) 343; l19B (1982) 231. [6] C. Angelini and R. Pazzi, Phys. Lett. 135B (1984) 473. [7] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. [8] W. Furmanski and R. Petronzio,Nucl. Phys. B195 (1982) 237.

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