Does F2 need a hard pomeron?

(1)

arXiv:hep-ph/0106307v2 20 Jul 2001

ULG-PNT-JRC-2001-1

Does

F

2

need a hard pomeron?

J.R. Cudell

1

and G. Soyez

2

Inst. de Physique, Bât. B5,Université deLiège, SartTilman,B4000 Liège,Belgium

Abstra t

WeshowthatthelatestHERAmeasurementsof

F

2

are ompatiblewiththeexisten e of a triple pole at

J

= 1

. Su h a stru ture also a ounts for soft forward hadroni

amplitudes,sothatthe introdu tion ofanewsingularityisnot ne essarytodes ribe

DIS data.

The latest data from HERA [1, 2, 3℄ have attained su h a level of pre ision that they

an onstrain quite stringently the possible singularity stru tures in the omplex

J

plane

whi h ontrol the

s

dependen e of soft hadroni amplitudes at

t = 0

and the

x

depen-den e of

F

2

. Traditionally, it has been assumed that soft hadroni amplitudes would be

onstrained by their analyti properties, i.e. by Regge theory, whereas DIS would be the

domainwhereperturbativeQCD ould betested,with anoverlap atonlyextremelysmall

x

. However, it has re ently been realised [4, 5℄ that there is no reason to limit the

ap-pli ation of Regge theory to su h a domain. Indeed, hadroni s attering amplitudes are

the ontinuation of

t

- hannel s attering amplitudes, and the simple stru tures predi ted

by Regge theory be ome valid on e the osine of the

t

- hannel s attering angle,

cos θ

t

,

be omessu iently large. One an infa t quantify this inthe hadron-hadron ase, asthe

forward s attering amplitudes (measured through

ρ

and

σ

tot

) are well tted for

√

s > 10

GeV. This orresponds, in the

pp

and

pp

ases, toa value of

cos θ

t

=

s

2m

2 p

≥ 57.

(1) In the

γ

∗

_p

ase, the total ross se tion,

σ

γ

∗

_p

=

4π

2 α

elm

Q

2 F

2 (x, Q

2 ),

(2) 1 JR.Cudellulg.a .be 2 G.Soyezulg.a .be

(2)

shouldbedes ribedby thesamesingularitiesforthesameregionof

cos θ

t

= ν/(m

p

√

Q

2 ) =

√

Q

2 _/(2m

p

x).

Hen e one would expe t toobservethe same singularitiesin

F

2

andin total hadroni ross se tions for values

x ≤

√

Q

2 100m

p

.

(3)

This isaratherlargeintervalwherethereis onsiderableoverlapwithstandard

pertur-bativeQCDevolution,hen eonewouldhopeforthetwodes riptionstobe ome ompatible

(butwearenotyetatastagetoshowthisresult). WhetheritmakessensetoextendRegge

ts to larger

x

[6, 7℄or tosmaller

s

[7℄remains debatable.

The problemin implementing su h a programis that, although the same singularities

haveto be present, their residues are not known, and must be

Q

2

-dependent. It is known

however that the

Q

2

dependen e must enter only in the residues, i.e. the

ν

dependen e

shouldnot hangewith

Q

2

. Astheee tive

ν

dependen edoesseemtovaryas

Q

2

in reases

atHERA, this an onlybeobtained by ombining several singularities,and writing

F

2 (

Q

2 2ν

, Q

2 ) =

X

i

g

i

(Q

2 )f

i

(ν)

(4)

with the

f

i

resultingfrom the singularities observed in ts tototal ross se tions [8℄. The

fa t that the residues depend on

Q

2

means that some singularities may remain hidden

in soft s attering amplitudes, and manifest themselves only at high

Q

2

. However, as the

values of

Q

2

observed now at HERA span the whole range from 0.045 to 30000 GeV

2

, it

seems unlikely that su h singularities would swit h ototally intotal ross se tions 3

.

The rstmodelproposingsu haglobaltisduetoDonna hieandLandsho[4℄andis

basedontheir previousmodel[10℄basedonasimple-polesingularity,whi ha ountsvery

su essfully not only for hadroni total ross se tions, but also for elasti and dira tive

s attering [10℄. The problem is that, at small

x

, this model predi ts a single singularity

in (4), and hen e the ee tive power of

x

annot hange with

Q

2

. This lead Donna hie

and Landshotopostulatethe existen eof ahiddensingularity, alledthe hard pomeron,

whi h is needed to reprodu e the rise of

F

2

at small

x

, but whi h is totally absent from

total hadroni ross se tions, with residues at least

10

6

times smaller than those of the

soft pomeron [8℄. It is rather hard to imagine how su h a er e singularity an totally

turn o in hadron-hadron s attering, as the hadrons will sometimes u tuate into small

systems of quarks, omparable to those produ ed by the o-shell photons. Furthermore,

the modelseems torequirethe softpomerontoturn oin

F

c

2

,whereas itisneeded in

J/ψ

produ tion. Althoughone annotprovethatthese featuresareex luded, theyseem rather

unnatural. Nevertheless, the model reprodu es quitewell the new data from HERA, and

an even be extended outsidethe area ofappli abilityof Regge theory [6℄.

3

Itishoweveralsopossibleto arguethatthephotonisspe ial,andthat t- hannelunitarityrelations,

fromwhi htheuniversalityofresiduesisderived[9℄,donotforma losedsystemforphotons,hen ethey

(3)

whi h work equally well, or better, than simple poles. In su h models, the rise of total

ross se tions isassumed to ome from adouble pole, or atriple pole, at

J = 1

.

In the rst ase [7℄, the t to soft amplitudes for es one to assume a large splittingof

the inter epts of leading meson traje tories, whi h means that su h traje tories must be

nonlinear. Furthermore,theleading

C = +1

ontributionbe omesnegativeatsu iently

small

s

,of theorderof10GeV,andfa torisationwould naivelyleadtonegativetotal ross

se tions for pro esses whi h ouple only to the leading traje tories. Although on e again

these points annotbeused torulethe modelout (as neitherfa torisationof doublepoles

nor the linearity of mesontraje tories an be proven), they make itseem unnatural.

The se ond andidate [12℄ does not have these drawba ks, and the des ription of total

ross se tions that it provides an even be extended down to

√

s = 5

GeV [11℄. Not

only does it t well both total ross se tions, and values of the

ρ

parameter for forward

hadroni amplitudes, but the out ome of su h a t seems the most natural: unitarity is

preserved, the various

C = +1

terms of the ross se tion remain positive, and the meson

traje tories are ompatible with ex hange degenera y. We shall show here that this t

extends naturally toreprodu eDIS data, atleast in the Regge region (3).

We shall adopt here the natural Regge variable

2ν

instead of

W

[7℄, as this makes a

sizable dieren ein the large-

x

region,and we shall assume that

F

2

an betted by

F

2 (

Q

2 2ν

, Q

2 ) = a(Q

2 ) log

2 2ν

2ν

0 (Q

2 )

!

+ c(Q

2 ) + d(Q

2 )(2ν)

−0.47

(5)

in the region (3). The rst two terms represent pomeron ex hange, and the third

a/f

ex hange. Wehavexed theinter ept ofthe latterfromtstohadroni amplitudes[8,11℄.

The rst test to he k whether su h a simple form has a han e to reprodu e the

data is to follow [4℄ and extra t the residues

a(Q

2 )

,

c(Q

2 )

,

d(Q

2 )

and the s ale

b(Q

2 ) ≡

log (2ν

0 (Q

2 ))

by ttingthe

ν

dependen e for ea h value of

Q

2

.

perdata pointof 0.674, omparedwith

1.00 forthe hardpomeronmodelof[4℄retted tothose data. Hen e itseems thata priori

the data are as ompatible with (5) as with the ts of [4, 6℄. The graphs represent the

oe ientsrespe tivelyof the

log

2

(2

ν

),

log

(2

ν

)and onstantterms. Wesee that thedata

do exhibit a suppression at small

Q

2 ,

and there is a strong orrelation between the three

terms, whi h allseem to have a similar behaviourwith

Q

2

(4)

(a) (b) Q 2 (GeV 2 ) a ( Q 2 ) 1000 100 10 1 0.1 0.05 0.04 0.03 0.02 0.01 0 Q 2 (GeV 2 ) 2 a ( Q 2 ) b ( Q 2 ) 1000 100 10 1 0.1 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ( ) Q 2 (GeV 2 ) a ( Q 2 ) b 2 ( Q 2 ) + ( Q 2 ) 1000 100 10 1 0.1 3 2.5 2 1.5 1 0.5 0

Figure1: the oe ients(a)oflog

2 (2v)

,(b)oflog

(2ν)

and ( )ofthe onstant term, extra ted fromthelatest HERAdata.

We an now pro eed to a parametrisation of the residues. We rst onsider the data of

refs. [1, 2℄, whi h extend to

Q

2 = 135

GeV

2

, as they are almost entirely in region (3),

together with data for the total

γp

ross se tion [13℄ at

√

_{s >}

5 GeV 4

. Apart fromthe fa t

that gauge invarian eimposesthat the formfa tors vanish linearly at

Q

2 → 0

forxed

ν

,

their formis undetermined. Wend that a good tis obtained by the following forms:

g(Q

2 ) = A

g

Q

2

1 1 + Q

2 /Q

2 g

!

ǫ

g

, g = a, c, d

(6) 4

Asthedatafrom5to10GeV havebigerrorbars,itmakeslittledieren ewhether wein ludethem

(5)

b(Q

2 ) = log

h

2ν

0 (Q

2 )

i

= b

0 + b

1 Q

2 Q

2 _{+ Q}

2 b

!

ǫ

_b

(7)

Q

2 ≤ 135

GeV

2 Q

2 ≤ 30000

ǫ

′

b

1.8551 .15402

Table 1: the values of the parameters orresponding to Eqs. (6, 7) for the

low-to-intermediate

Q

2

t, and those of the global t to region (3), orresponding to Eqs.

(6,8). p s (GeV) tot (m b) 100 10 0.2 0.18 0.16 0.14 0.12 0.1

Figure2: t to the total rossse tion datafor

√

(6)

the t smoothly reprodu es the value of

ν

0

2

. Whether one wants to go

beyond this result isreally a matterof taste. As we have explained,it is possible to have

hidden singularities, whi hwouldmanifest themselvesonlyas

Q

2

be omeslarge enough.

From aperturbative QCDpointof view, the presentt an beseen asastartingpoint for

F

2 (x, Q

2 )

, onsistingof onstants,

log x

and

log

2 x

terms. 5

Perturbativeevolutionwillthen

5

Onemust note howeverthat wehave nottried to gobeyond the Regge region, and hen ethe limit

x → 1

isnot orre t,as

F

(7)

produ efurther

log x

terms,whi hwould behiddenbelowthe startingevolutions ale. As

singularitiesofhadroni amplitudes annoto uratarbitrarypla es,thelatterwouldhave

to be a physi al parameter, and the present t shows that it should be of order

Q

0 = 10

GeV [14℄.

Itmaybeaworthwhileexer isehoweverto he k whetherthissingularitystru ture an

be extended to the full Regge region, and to onsider a t to the full datasetof DIS [15℄.

The reason is that in this ase there is onsiderable overlap between the DGLAP region

(

Q

2

large,

log Q

2 ≫ − log x

) and the Regge region (3), and hen e one should be able to

understand some features of the pomeronthrough perturbative means. In this region, to

a hieve values of

χ

2

omparabletothose of[6,7℄,we needinfa t tomodifythe formused

for the s ale

ν

0 (Q

2 )

,and introdu ea logarithmin itsexpression:

b(Q

2 ) = b

0 + b

′

1 "

log

1 +

Q

2 Q

′

2 b

!#

ǫ

′

_b

.

(8)

Su h a form allows us to extend the t to the full Regge region (3), and produ es a

reasonable

χ

2 /dof

: we obtain1411 for 1166 points(in luding 21pointsforthe total ross

se tion above

√

In on lusion, we see that several s enarios ompatible with Regge theory are possible

todes ribestru ture fun tions. All ofthem sharethe hara teristi that oneneeds several

omponent to des ribe DIS and soft s attering at the same time. We believe that the

present parametrisationshows thatno unexpe ted behaviour is needed to reprodu e DIS,

and that the pomeron may wellbe asingle obje t, onne ting all regions of

Q

2

smoothly,

and exhibiting the same singularitiesin DIS and in soft s attering. Howto obtain su h a

simpleforminthe regionofoverlap between perturbativeQCDand Reggetheory remains

anopen question.

A knowledgements

We thank P.V. Landsho for fruitful ex hanges, and E. Martynov for sharing his results

with us, for numerous dis ussions, and for orre ting amistake ina preliminaryversion.

6

Notethatthetof[6℄givea

χ

2

(8)

0

0.5

1

1.5

0.045

0.065

0.085

0.11

0

0.5

1

1.5

0.15

0.2

0.25

0.3

0

0.5

1

1.5

0.4

0.5

0.6

0.65

0

0.5

1

1.5

0.85

0.9

1.2

1.3

0

0.5

1

1.5

1.9

2.0

2.5

0

0.5

1

1.5 1e-06

0.0001

0.01

3.0 1e-06

0.0001

0.01

3.5 1e-06

0.0001

0.01

4.5 1e-06

0.0001

0.01

5.0

Figure 4 (a): Resultof a global t to all the data in the Regge region, for

0.045 ≤

Q

2 _{≤ 5.0}

GeV

2

.

F

2 (x, Q

2 )

is shown as a fun tion of

x

,for ea h

Q

2

value indi ated

(inGeV

2

(9)

0

0.5

1

1.5

6.0

6.5

7.5

8.5

0

0.5

1

1.5

9.0

10.0

12.0

15.0

0

0.5

1

1.5

17.0

18.0

20.0

22.0

0

0.5

1

1.5

25.0

27.0

30.0

35.0

0

0.5

1

1.5

45.0

50.0

60.0

65.0

0

0.5

1

1.5 1e-06

0.0001

0.01

70.0 1e-06

0.0001

0.01

80.0 1e-06

0.0001

0.01

90.0 1e-06

0.0001

0.01

120.0

Figure4(b): Resultofaglobaltto allthedataintheReggeregion,for

6.0 ≤ Q

2 ≤

120.0

GeV

2

.

F

2 (x, Q

2 )

is shown as a fun tion of

x

, for ea h

Q

2

value indi ated (in

GeV

2

(10)

0

0.5

1

1.5

125.0

150.0

200.0

240.0

0

0.5

1

1.5

250.0

300.0

350.0

400.0

0

0.5

1

1.5

450.0

480.0

500.0

650.0

0

0.5

1

1.5

800.0 1000.0

1200.0

1500.0

0

0.5

1

1.5 1600.0

2000.0

3000.0

5000.0

0

0.5

1

1.5 1e-06

0.0001

0.01 8000

1e-06

0.0001

0.01 12000

1e-06

0.0001

0.01 20000

1e-06

0.0001

0.01 30000

Figure 4 ( ): Result of a global t to all the data in the Reggeregion, for

125.0 ≤

Q

2 _{≤ 30000.0}

GeV

2

.

F

2 (x, Q

2 )

isshownasafun tionof

x

,forea h

Q

2

valueindi ated

(inGeV

2

(11)

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