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Fracture functions and Jet Calculus
5 October 1998
Physics Letters B 439 Ž1998. 382–388
Fracture functions and Jet Calculus Gianni Camici a , Massimiliano Grazzini b, Luca Trentadue b
b
a Dipartimento di Fisica,UniÕersita` di Firenze, INFN Sezione di Firenze, 50125 Firenze, Italy Dipartimento di Fisica, UniÕersita` di Parma, INFN Gruppo Collegato di Parma, 43100 Parma, Italy
Received 7 April 1998; revised 17 July 1998 Editor: R. Gatto
Abstract By using Jet Calculus as a consistent framework to describe multiparton dynamics we explain the peculiar evolution equation of fracture functions by means of the recently introduced extended fracture functions. q 1998 Published by Elsevier Science B.V. All rights reserved. PACS: 13.85.Ni
1. Introduction Fracture functions w1x have been introduced to interpret within the framework of perturbative QCD semi-inclusive deep inelastic processes in the target fragmentation region. The formulation of these processes by means of fracture functions does allow to extend the interpretation in terms of QCD-improved parton model to the complementary region of target fragmentation. This implies a description of deep inelastic processes in terms of the novel fracture functions in addition to the usual factorizable structure and fragmentation functions convoluted with hard point-like cross sections. The introduction of these new objects requires however the formulation of an additional factorization hypothesis whose validity has been recently argued on the basis of the cut vertex formalism together with the use of infrared power counting.
These results have been obtained within the theoretical framework of Ž f 3 . 6 field theory w2,3x and have been later confirmed in QCD in Ref. w4x. In order to show that the new factorization hypothesis is correct, in Ref. w2x new objects have been defined called extended fracture functions, which depend also on the momentum transfer t between the incoming and outgoing hadron. By an explicit oneloop calculation it has been verified that these object do factorize and it has been also observed that they show a new logarithmic dependence on the ratio of the scales Q 2 and t w5x. As a result extended fracture functions obey an evolution equation different from the one followed by ordinary fracture functions. The aim of this note is to show in a simple and direct way that extended and ordinary fracture functions are closely connected. As a consequence the corresponding evolution equation do follow one from the other. We work within the framework of Jet
0370-2693r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 4 0 - 5
G. Camici et al.r Physics Letters B 439 (1998) 382–388
Calculus w6x by applying the corresponding rules to the evolution equations. The set of Jet Calculus rules allows the interpretation in terms of QCD-improved parton model of the results obtained by using a different approach w2x. By using this method we can give in fact a definition of the extended fracture function in the region where t is a perturbative scale aS Ž t .
Q 1.. As a consequence a DGLAP evolution equation for extended fracture functions follows and the inhomogeneous evolution equation for the ordinary fracture function is recovered. Ž
2p
2. Evolution pattern In a deep inelastic semi-inclusive reaction, a hadron A with momentum p is struck by a far off-shell spacelike photon with momentum q and a hadron AX with momentum pX is inclusively observed in the final state. Let us define as usual Q 2 s yq 2 x s
Q2 2 pq
Ž 1.
and choose a frame in which p s Ž pq, py,0. with pq4 py and pq , pq qy. As far as we keep away from the target fragmentation region the cross section factorizes as follows w7x
sJ s
dx X dz X
Hx
X
zX
FAi Ž x X ,Q 2 . sˆi j Ž xrx X , zrz ˜ ˜X ,Q 2 .
=DAjX Ž z˜X ,Q 2 .
Ž 2.
where FAi Ž x,Q 2 . and DAjX Ž z,Q ˜ 2 . are the structure and fragmentation function respectively, sˆi j Ž x, z,Q ˜ 2 . is the hard semi-inclusive cross section and we have X defined z˜ s ppXrpq , py rqy w7x. Eq. Ž2. expresses the fact that, as long as the produced hadron has a large transverse momentum pXT2 ; Q 2 Ži.e. z˜ is finite. it can be thought as a product of the fragmentation of the active parton. However in the last few years, especially after the appearance of diffractive deep inelastic events at HERA, particular attention has been payed to hadron production in the target fragmentation region Ži.e. the region z˜ ™ 0., where Eq. Ž2. fails. In Ref. w1x it has
383
been proposed for target fragmentation the additional factorized term dx X i sT s M X Ž x X , z ,Q 2 . sˆi Ž xrx X ,Q 2 . Ž 3. xX A A X where z s pX qrpq , pq rpq represent the momentum fraction of the hadron AX with respect to A. Here MAi AX Ž x, z,Q 2 . is the fracture function, giving the probability of finding a parton i with momentum fraction x in the hadron A while another hadron AX with momentum fraction z is detected. The fracture function is expected to satisfy an inhomogeneous evolution equation w1x, and this fact has been verified at one-loop level in Ref. w8x. Let us consider the case in which the momentum transfer t s yŽ p y pX . 2 < Q 2 is also measured. The current and target fragmentation contributions are in this case dx X dz X i X 2 sJ s F Ž x ,Q . xX zX A
H
H
=sˆi j Ž xrx X , zrx X z X ,tx Xrz X ,Q 2 . DAjX Ž z X ,Q 2 . Ž 4 . and
sT s
dx X
Hx
X
MAi AX Ž x X , z ,t ,Q 2 . sˆi Ž xrx X ,Q 2 .
Ž 5.
The function MA,i AX Ž x, z,t,Q 2 . is the extended fracture function. In Ref. w2x it has been shown that this object can be defined just in terms of a new cut vertex. Whereas MA,i AX Ž x, z,Q 2 . is expected to satisfy an inhomogeneous evolution equation w1x, the extended fracture function obeys a simple DGLAP evolution equation w2x. This facts can be understood by making the following observations. In the region L2Q C D < t < Q 2 the hard semi-inclusive cross section sˆi j in Eq. Ž4. develops large log Q 2rt corrections which have to be resummed w5x. Instead of absorbing these logs in sˆi j we choose to move them in the extended fracture function which, by using Jet Calculus, can be defined in the perturbative region of t as MAj , AX Ž x , z ,t ,Q 2 . s
aS Ž t . 2p t
HF Ž w,t . dw Pˆ i A
i
kl
Ž u . du
=Dl , AX Ž j ,t . d j Ekj Ž r ,t ,Q 2 . dr =d Ž x y ruw . d Ž z y w Ž 1 y u . j .
G. Camici et al.r Physics Letters B 439 (1998) 382–388
384
s
aS Ž t . 2p t =Pˆik l
s
x
=Pˆik l
w y xrr D l , AX
rw
1yz
Hx r
dr
FAi Ž w,t .
Q 2 and t in terms of logarithms of the form log Qt and at first order in a S has the following expression
z
Eij Ž x ,t ,Q 2 . , d i jd Ž 1 y x . q
w y xrr
,t Ekj Ž r ,t ,Q 2 .
/
dw
1
D l , AX
z wyr
i A
,t Ekj Ž xrr ,t ,Q 2 .
/
Ž 6.
where the integration limits have been obtained implementing momentum conservation. We work within the leading logarithmic approximation. Here Pi j Ž u. and Pˆi jk Ž u. are regularized and real Altarelli-Parisi vertices w6 x respectively, and the function Ekj Ž x,Q 02 ,Q 2 . is the evolution kernel from Q 02 to Q 2 , which obeys the DGLAP evolution equation:
E Q2
E Q2
Eij Ž x ,Q02 ,Q 2 . s
aS Ž Q 2p
.
1 du
Hx
2p
Pi j Ž x . log
Q2 t
.
In this way we have absorbed these large logarithms in the definition of MA,i AX Ž x, z,t,Q 2 . which in the perturbative region of t is resolved in the convolution of the objects contained in the dotted box in Fig 1. From Eq. Ž6. it appears that the Q 2 dependence of the extended fracture function is completely given by the evolution kernel Eik Ž x,Q 02 ,Q 2 . and therefore the evolution equation is E Q2 M j X Ž x , z ,t ,Q 2 . E Q2 A, A s
2
aS
Ž 8.
F Ž w,t . H r zqr w Ž w y r .
ž / ž w
2
dr
ž / ž
aS Ž t . 2p t
dw
H rw
u
aS Ž Q 2 . 2p
Pkj Ž u .
=Eik Ž xru,Q02 ,Q 2 . .
du
1
Hx 1yz
u
Pi j Ž u . MAi , AX Ž xru, z ,t ,Q 2 .
Ž 9. Ž 7.
Eq. Ž6. is represented in Fig. 1. The evolution kernel Žthe black blob in Fig. 1. resums the dependence on
at least in the perturbative region of t, i.e. t R L2QC D . Therefore the anomalous dimension associated to the extended fracture function is the same that controls the evolution of ordinary structure functions. Let us suppose now that, as argued in Ref. w2x, the evolution Eq. Ž9. applies also within the small t region and define the integrated fracture function as an integral over t up to a Q 2-dependent cut-off of order Q 2 , say e Q 2 , with e - 1 MAj , AX Ž x , z ,Q 2 . s
2
H0e Q dt M
j X A, A
Ž x , z ,t ,Q 2 . .
Ž 10 .
By taking the logarithmic derivative of Eq. Ž10. we have that E Q2 M j X Ž x , z ,Q 2 . E Q2 A, A E e Q2 s dt Q 2 M j X Ž x , z ,t ,Q 2 . E Q2 A, A 0
H
q e Q 2 MAj , AX Ž x , z , e Q 2 ,Q 2 . s
aS Ž Q 2 . 2p
2
1
H0e Q dtH x
1yz
du u
=Pi j Ž u . MAi , AX Ž xru, z ,t ,Q 2 . Fig. 1. Extended fracture function in the perturbative region of t.
q e Q 2 MAj , AX Ž x , z , e Q 2 ,Q 2 .
G. Camici et al.r Physics Letters B 439 (1998) 382–388
s
aS Ž Q 2 . 2p
du
1
Hx
Pi j Ž u . MAi , AX Ž xru, z ,Q 2 .
u
1yz
Ž 11 .
We see appearing an inhomogeneous term in the evolution equation of MA,j AX Ž x, z,Q 2 . which arises from the Q 2 dependence of the upper integration limit. This inhomogeneous term depends on the value of MA,j AX Ž x, z,t,Q 2 . in the perturbative region of t where Eq. Ž6. holds. By using the boundary condition Ekj Ž x ,Q 2 ,Q 2 . s d kj d Ž 1 y x .
e Q 2 MAj , AX Ž x , z , e Q 2 ,Q 2 . s
2p =Pˆi jl
s
r
w
2
.
2p
s
Hx
x
Dl , AX
aS Ž Q 2 . 2p
wyr
x
Hx
=Pˆi jl Ž u . D l , AX
z wyx
du
xqz
Ž w,Q .
,Q 2
zu x Ž 1 y u.
2
.
/
u
FAi Ž xru,Q 2 .
,Q 2 .
E EQ
s
2
1yz
m
MAj , AX Ž x , z ,Q 2 .
2p
1
Hx 1yz
X
dxx n MAj , AX Ž x , z ,Q 2 . s Mmj,nA A Ž Q 2 . .
Ž 15 . One can verify that Eq. Ž6. becomes aS Ž t . k l i i Mmj n Ž t ,Q 2 . s P F Ž t . Dmk Ž t . Enjl Ž t ,Q 2 . 2p t n m mqn Ž 16 .
Pmk nl i s
1
H0 du u
/
du u
=Pi j Ž u . MAi , AX Ž xru, z ,Q 2 .
m
n Ž 1 y u . Pˆil k Ž u .
Ž 17 .
Ž 13 .
1
n
j l
H0 du u E Ž u,t ,Q
2
..
Ž 18 .
The t evolution equation contains several terms: the first comes from the canonical scale dependence of the extended fracture function, the second from the scale dependence of a S , and the third from the scale dependence of the evolution kernel. There are then two inhomogeneous terms which follow from the t dependence of structure and fragmentation functions respectively. By explicitly deriving Eq. Ž16. with respect to t we get E j t M Ž t ,Q 2 . E t mn aS Ž t . j p s y d j p Ž1 q b0 aS Ž t . . q An 2p
ž
/
= Mmpn a S2
2
Ž t ,Q .
q
aS Ž Q 2 .
/
and
By substituting Eq. Ž13. in Eq. Ž11. the evolution equation for MA,j AX Ž x, z,Q 2 . appears to be Q2
1
H0 dzz H0
Enjl Ž t ,Q 2 . s
i A
u x Ž 1 y u.
ž
Hx
ž
2
/
Hzqx w Ž w y x . F Ž w,Q Dl , AX
FAi
,Q 2 d Ž xrr y 1 .
dw
1
ž / ž w
Hzqr w Ž w y r . z
du
F i Ž xru,Q 2 . x Ž 1 y u. A zu =Pˆi jl Ž u . D l , AX ,Q 2 , Ž 14 . x Ž 1 y u. that is exactly the one proposed in Ref. w1x. Furthermore the perturbative definition Ž6. can be used to derive the evolution with t of the extended fracture function. In order to make the notation less cumbersome, it is convenient to define q
where we have defined w6x
dw
1
r
ž / ž
aS Ž Q
=Pˆi jl
dr
1yz
xqz
Ž 12 .
on Eq. Ž6. we get for the inhomogeneous term, up to log e corrections,
aS Ž Q 2 .
x
aS Ž Q 2 . 2p
q e Q 2 MAj , AX Ž x , z , e Q 2 ,Q 2 . .
385
Ž t. 2
Ž 2p . t =Enjl
2
p p Pnkml i Aimqn Fmqn Ž t . Dmk Ž t .
Ž t ,Q . q
a S2 Ž t . 2 Ž 2p . t
= A kmp Dmp Ž t . Enjl Ž t ,Q 2 .
i Pnkml i Fmqn Ž t.
Ž 19 .
G. Camici et al.r Physics Letters B 439 (1998) 382–388
386
where A nji s
1
n
j
H0 du u P Ž u . .
Ž 20 .
i
We conclude this section by stressing that one can define the fracture function in a more general way as an integral up to an arbitrary Q 2-dependent integration limit MAj , AX Ž x , z ,Q 2 . s
2
H0t ŽQ .dt M
j X A, A
2
Ž x , z ,t ,Q 2 .
Ž 21 .
One could even define a more general fracture function introducing also a lower integration limit t 1Ž Q 2 . MAj , AX Ž x , z ,Q 2 . s
2
Ht tŽQŽQ. .dt M 2
2
j X A, A
Ž x , z ,t ,Q 2 . . Ž 24.
1
and the equation would have an additional inhomogeneous term taking into account the Q 2 dependence of the lower integration limit
E Q2
In this case the evolution equation reads
E Q2
Mmj n Ž Q 2 .
E Q2
MAj , AX Ž x , z ,Q 2 .
E Q2
s
s
aS Ž Q 2 . 2p
u
1yz
aS Ž t2 Ž Q
q
2
a S Ž t 2 Ž Q 2 . . Q 2 t 2X Ž Q 2 .
..
2
X
Q t2 Ž Q t2 Ž Q
2
2
.
.
du
rqz
r Ž 1 y u.
=Pˆik l Ž u . D l , AX
1yz
Hx
ž
r Ž 1 y u.
i Pnkml i Fmqn Ž t2 Ž Q 2 . .
dr r
y
FAi Ž rru,t 2 Ž Q 2 . . zu
t2 Ž Q 2 .
=Dmk Ž t 2 Ž Q 2 . . Enjl Ž t 2 Ž Q 2 . ,Q 2 .
a S Ž t 1 Ž Q 2 . . Q 2 t 1X Ž Q 2 . 2p
r
Hr
A nj p Mmp n Ž Q 2 .
2p
2p
=
2p
du
1
Hx
=Pi j Ž u . MAi , AX Ž xru, z ,Q 2 . q
aS Ž Q 2 .
,t 2 Ž Q 2 .
=Ekj Ž xrr ,t 2 Ž Q 2 . ,Q 2 .
t1 Ž Q 2 .
i Pnkml i Fmqn Ž t1 Ž Q 2 . .
=Dmk Ž t 1 Ž Q 2 . . Enjl Ž t 1 Ž Q 2 . ,Q 2 . .
/ Ž 22 .
Ž 25 .
Phenomenologically these two cases would correspond to the production of hadrons within the target fragmentation region with transverse momentum below t 2 Ž Q 2 . and between t 1Ž Q 2 . and t 2 Ž Q 2 . respectively.
or, by taking double moments,
E Q2
E Q2
s
Mmj n Ž Q 2 .
aS Ž Q 2 . 2p q
3. Sum rules
A nj p Mmp n Ž Q 2 .
a S Ž t 2 Ž Q 2 . . Q 2 t 2X Ž Q 2 . 2p
t2 Ž Q 2 .
In Ref. w1x it was shown that fracture functions obey the following momentum sum rule i Pnkml i Fmqn Ž t2 Ž Q 2 . .
Ý Hdz z MAj , A Ž x , z ,Q 2 . s Ž 1 y x . FAj Ž x ,Q 2 . Ž 26. X
X
A
=Dmk Ž t 2 Ž Q 2 . . Enjl Ž t 2 Ž Q 2 . ,Q 2 . .
Ž 23 .
In the case in which t 2 Ž Q 2 . s e Q 2 the evolution Eq. Ž14. is of course recovered.
which accounts for momentum conservation in the s-channel. In this section we want investigate if a momentum sum rule holds also in the t-channel.
G. Camici et al.r Physics Letters B 439 (1998) 382–388
Taking moments of Eq. Ž14. with m s 0, n s 1 and summing over j we get w9x
E Q2
EQ
s
X
2
Ý M01j, A A Ž Q 2 . j
aS Ž Q 2 . 2p
F1i , A Ž Q 2 .
X
1
H0 du u P
i
k
Ž 1 y u . D 0k , A Ž Q 2 . Ž 27 .
where we made use of the well-known property of the splitting function w6x
Ý A1ji s 0
Ž 28 .
j
and of the relation 1
H0 du Ž 1 y u . Ý Pˆ
jk
i
1
Ž u . s H du Ž 1 y u . Pi j Ž u . . 0
k
Ž 29 . Ž27. it appears that the derivative of From Eq. X j, A A Ž 2 . Ý j M01 Q receives contribution from the inhomogeneous term in the evolution equation. This fact suggests that the sum rule could be violated because at values of t of order Q 2 the current contribution becomes important and the two production mechanisms cannot be disentangled. As far as the extended fracture function is concerned, since MA,j AX Ž x, z,t,Q 2 . obeys the DGLAP evolution equation, it follows that
E Q2
EQ
1yz
2
ÝH j
0
x dx MAj , AX Ž x , z ,t ,Q 2 . s 0
Ž 30 .
and momentum conservation in the t channel is recovered. In the perturbative region of t, by using Eq. Ž6., we find X
Ý M01j, A A Ž t ,Q 2 . j
s
aS Ž t . 2p t
F1i , A Ž t .
X
1
H0 du uP
i
k
Ž 1 y u . D 0k , A Ž t . . Ž 31 .
4. Summary In this note we showed that a formulation of perturbative processes in the target fragmentation
387
region can be given by using the formalism of Jet Calculus w6x. A perturbative evaluation in terms of evolution equation and anomalous dimensions of extended fracture function is consistent with the evolution equations proposed in Ref. w1x and w2x for ordinary and extended fracture functions respectively. We showed that the inhomogeneous term in Eq. Ž14. precisely stems from the integration over momentum transfer t. Moreover in this formalism the extension to nextto-leading order seems quite natural and suggests that Eqs. Ž9. and Ž14. keep the same structure in the two-loops approximation, analogously to what happens for ordinary structure and fragmentation functions w10x. In the region L2QC D < t < Q 2 the hard semi-inclusive cross section develops large log Q 2rt corrections which need to be resummed. Our approach is based on the idea of absorbing such corrections in the definition of MA,i AX Ž x, z,t,Q 2 ., namely, logQ 2rt are fully contained in the evolution kernel Eik Ž x,t,Q 2 .. These logs, which can be treated with standard renormalization group techniques, are new and potentially useful to understand the dynamics of hadro-production in the target fragmentation region. These logs could affect significantly observables like multiplicities and particle distributions and we believe that a phenomenological study of the role of these corrections would be useful. As originally proposed in Ref. w1x and in Ref. w11x, there now seems to be a widespread consensus that the Q 2 evolution in the diffractive regime can be described in terms of the perturbative QCD formalism w4,12,13x. The data taken by H1 collaboration w14x, showing an evidence of a consistent logarithmic scaling violation in diffractive channels have further stressed that such an interpretation is not far from the experimental observation. All this is consistent with a description in terms of fracture functions. Our results agree with such an expectation, although in the limit z ™ 1 our formulae will show the appearance of large logŽ1 y z . factors which should be resummed as well. For the treatment of such a singular region we believe an approach can be used close to the one followed in the inclusive case. As a concluding remark we want to stress that the results presented here completely agree with those of
388
G. Camici et al.r Physics Letters B 439 (1998) 382–388
Ref. w2x, obtained within a quite different framework, namely, that of a generalized cut vertex expansion which, as a generalization of Operator Product Expansion, appears to be founded on a firmer theoretical standpoint. Once more the equivalence of parton-like and OPE-like approaches is verified.
w3x w4x w5x w6x w7x w8x w9x
Acknowledgements Many of the arguments covered in this letter originate from discussions with G. Veneziano. We also thank G. Veneziano for useful comments, and S. Catani, D. Graudenz and J. Kodaira for conversations. One of us ŽM.G.. would like to thank the Department of Physics of the university of Florence for the hospitality at various stages of this work.
w10x w11x w12x
References w13x w1x L. Trentadue, G. Veneziano, Phys. Lett. B 323 Ž1994. 201. w2x M. Grazzini, L. Trentadue, G. Veneziano, Nucl. Phys. B 519 Ž1998. 394.
w14x
M. Grazzini, Phys. Rev. D 57 Ž1998. 4352. J.C. Collins, Phys. Rev. D 57 Ž1998. 3051. M. Grazzini, Nucl. Phys. B 518 Ž1998. 303. K. Konishi, A. Ukawa, G. Veneziano, Nucl. Phys. B 157 Ž1979. 45. G. Altarelli, R.K. Ellis, G. Martinelli, S.Y. Pi, Nucl. Phys. B 160 Ž1979. 301. D. Graudenz, Nucl. Phys. B 432 Ž1994. 351. L. Trentadue, Proceedings of QCD Euroconference 96 ŽMontpellier, France July 1996. published in Nucl. Phys. Proc. Suppl. 54A Ž1996. 176; see also D. Graudenz, Fortsch. Phys. 45 Ž1997. 629. J. Kalinowski, K. Konishi, P.N. Scharbach, T.R. Taylor, Nucl. Phys. B 181 Ž1981. 221, 253. A. Berera, D.E. Soper, Phys. Rev. D 53 Ž1996. 6162. Z. Kunszt, W.J. Stirling, talk given at Durham Workshop on HERA Physics: Proton, Photon, and Pomeron Structure, Durham, England, 17-23 September 1995, and at International Workshop on Deep Inelastic Scattering and Related Phenomena ŽRome, Italy, April 1996., hep-phr9609245; Z. Kunszt, talk given at Ringberg Workshop on New Trends in HERA Physics ŽRingberg Castle, Germany, May 1997. hepphr9710211. D. De Florian, R. Sassot, hep-phr9710205, to be published in Proceedings of the Madrid low-x Workshop ŽMiraflores de la Sierra, Spain, June 1997.. H1 Collaboration, Z. Phys. C 76 Ž1997. 613.
Physics Letters B 439 Ž1998. 382–388
Fracture functions and Jet Calculus Gianni Camici a , Massimiliano Grazzini b, Luca Trentadue b
b
a Dipartimento di Fisica,UniÕersita` di Firenze, INFN Sezione di Firenze, 50125 Firenze, Italy Dipartimento di Fisica, UniÕersita` di Parma, INFN Gruppo Collegato di Parma, 43100 Parma, Italy
Received 7 April 1998; revised 17 July 1998 Editor: R. Gatto
Abstract By using Jet Calculus as a consistent framework to describe multiparton dynamics we explain the peculiar evolution equation of fracture functions by means of the recently introduced extended fracture functions. q 1998 Published by Elsevier Science B.V. All rights reserved. PACS: 13.85.Ni
1. Introduction Fracture functions w1x have been introduced to interpret within the framework of perturbative QCD semi-inclusive deep inelastic processes in the target fragmentation region. The formulation of these processes by means of fracture functions does allow to extend the interpretation in terms of QCD-improved parton model to the complementary region of target fragmentation. This implies a description of deep inelastic processes in terms of the novel fracture functions in addition to the usual factorizable structure and fragmentation functions convoluted with hard point-like cross sections. The introduction of these new objects requires however the formulation of an additional factorization hypothesis whose validity has been recently argued on the basis of the cut vertex formalism together with the use of infrared power counting.
These results have been obtained within the theoretical framework of Ž f 3 . 6 field theory w2,3x and have been later confirmed in QCD in Ref. w4x. In order to show that the new factorization hypothesis is correct, in Ref. w2x new objects have been defined called extended fracture functions, which depend also on the momentum transfer t between the incoming and outgoing hadron. By an explicit oneloop calculation it has been verified that these object do factorize and it has been also observed that they show a new logarithmic dependence on the ratio of the scales Q 2 and t w5x. As a result extended fracture functions obey an evolution equation different from the one followed by ordinary fracture functions. The aim of this note is to show in a simple and direct way that extended and ordinary fracture functions are closely connected. As a consequence the corresponding evolution equation do follow one from the other. We work within the framework of Jet
0370-2693r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 4 0 - 5
G. Camici et al.r Physics Letters B 439 (1998) 382–388
Calculus w6x by applying the corresponding rules to the evolution equations. The set of Jet Calculus rules allows the interpretation in terms of QCD-improved parton model of the results obtained by using a different approach w2x. By using this method we can give in fact a definition of the extended fracture function in the region where t is a perturbative scale aS Ž t .
Q 1.. As a consequence a DGLAP evolution equation for extended fracture functions follows and the inhomogeneous evolution equation for the ordinary fracture function is recovered. Ž
2p
2. Evolution pattern In a deep inelastic semi-inclusive reaction, a hadron A with momentum p is struck by a far off-shell spacelike photon with momentum q and a hadron AX with momentum pX is inclusively observed in the final state. Let us define as usual Q 2 s yq 2 x s
Q2 2 pq
Ž 1.
and choose a frame in which p s Ž pq, py,0. with pq4 py and pq , pq qy. As far as we keep away from the target fragmentation region the cross section factorizes as follows w7x
sJ s
dx X dz X
Hx
X
zX
FAi Ž x X ,Q 2 . sˆi j Ž xrx X , zrz ˜ ˜X ,Q 2 .
=DAjX Ž z˜X ,Q 2 .
Ž 2.
where FAi Ž x,Q 2 . and DAjX Ž z,Q ˜ 2 . are the structure and fragmentation function respectively, sˆi j Ž x, z,Q ˜ 2 . is the hard semi-inclusive cross section and we have X defined z˜ s ppXrpq , py rqy w7x. Eq. Ž2. expresses the fact that, as long as the produced hadron has a large transverse momentum pXT2 ; Q 2 Ži.e. z˜ is finite. it can be thought as a product of the fragmentation of the active parton. However in the last few years, especially after the appearance of diffractive deep inelastic events at HERA, particular attention has been payed to hadron production in the target fragmentation region Ži.e. the region z˜ ™ 0., where Eq. Ž2. fails. In Ref. w1x it has
383
been proposed for target fragmentation the additional factorized term dx X i sT s M X Ž x X , z ,Q 2 . sˆi Ž xrx X ,Q 2 . Ž 3. xX A A X where z s pX qrpq , pq rpq represent the momentum fraction of the hadron AX with respect to A. Here MAi AX Ž x, z,Q 2 . is the fracture function, giving the probability of finding a parton i with momentum fraction x in the hadron A while another hadron AX with momentum fraction z is detected. The fracture function is expected to satisfy an inhomogeneous evolution equation w1x, and this fact has been verified at one-loop level in Ref. w8x. Let us consider the case in which the momentum transfer t s yŽ p y pX . 2 < Q 2 is also measured. The current and target fragmentation contributions are in this case dx X dz X i X 2 sJ s F Ž x ,Q . xX zX A
H
H
=sˆi j Ž xrx X , zrx X z X ,tx Xrz X ,Q 2 . DAjX Ž z X ,Q 2 . Ž 4 . and
sT s
dx X
Hx
X
MAi AX Ž x X , z ,t ,Q 2 . sˆi Ž xrx X ,Q 2 .
Ž 5.
The function MA,i AX Ž x, z,t,Q 2 . is the extended fracture function. In Ref. w2x it has been shown that this object can be defined just in terms of a new cut vertex. Whereas MA,i AX Ž x, z,Q 2 . is expected to satisfy an inhomogeneous evolution equation w1x, the extended fracture function obeys a simple DGLAP evolution equation w2x. This facts can be understood by making the following observations. In the region L2Q C D < t < Q 2 the hard semi-inclusive cross section sˆi j in Eq. Ž4. develops large log Q 2rt corrections which have to be resummed w5x. Instead of absorbing these logs in sˆi j we choose to move them in the extended fracture function which, by using Jet Calculus, can be defined in the perturbative region of t as MAj , AX Ž x , z ,t ,Q 2 . s
aS Ž t . 2p t
HF Ž w,t . dw Pˆ i A
i
kl
Ž u . du
=Dl , AX Ž j ,t . d j Ekj Ž r ,t ,Q 2 . dr =d Ž x y ruw . d Ž z y w Ž 1 y u . j .
G. Camici et al.r Physics Letters B 439 (1998) 382–388
384
s
aS Ž t . 2p t =Pˆik l
s
x
=Pˆik l
w y xrr D l , AX
rw
1yz
Hx r
dr
FAi Ž w,t .
Q 2 and t in terms of logarithms of the form log Qt and at first order in a S has the following expression
z
Eij Ž x ,t ,Q 2 . , d i jd Ž 1 y x . q
w y xrr
,t Ekj Ž r ,t ,Q 2 .
/
dw
1
D l , AX
z wyr
i A
,t Ekj Ž xrr ,t ,Q 2 .
/
Ž 6.
where the integration limits have been obtained implementing momentum conservation. We work within the leading logarithmic approximation. Here Pi j Ž u. and Pˆi jk Ž u. are regularized and real Altarelli-Parisi vertices w6 x respectively, and the function Ekj Ž x,Q 02 ,Q 2 . is the evolution kernel from Q 02 to Q 2 , which obeys the DGLAP evolution equation:
E Q2
E Q2
Eij Ž x ,Q02 ,Q 2 . s
aS Ž Q 2p
.
1 du
Hx
2p
Pi j Ž x . log
Q2 t
.
In this way we have absorbed these large logarithms in the definition of MA,i AX Ž x, z,t,Q 2 . which in the perturbative region of t is resolved in the convolution of the objects contained in the dotted box in Fig 1. From Eq. Ž6. it appears that the Q 2 dependence of the extended fracture function is completely given by the evolution kernel Eik Ž x,Q 02 ,Q 2 . and therefore the evolution equation is E Q2 M j X Ž x , z ,t ,Q 2 . E Q2 A, A s
2
aS
Ž 8.
F Ž w,t . H r zqr w Ž w y r .
ž / ž w
2
dr
ž / ž
aS Ž t . 2p t
dw
H rw
u
aS Ž Q 2 . 2p
Pkj Ž u .
=Eik Ž xru,Q02 ,Q 2 . .
du
1
Hx 1yz
u
Pi j Ž u . MAi , AX Ž xru, z ,t ,Q 2 .
Ž 9. Ž 7.
Eq. Ž6. is represented in Fig. 1. The evolution kernel Žthe black blob in Fig. 1. resums the dependence on
at least in the perturbative region of t, i.e. t R L2QC D . Therefore the anomalous dimension associated to the extended fracture function is the same that controls the evolution of ordinary structure functions. Let us suppose now that, as argued in Ref. w2x, the evolution Eq. Ž9. applies also within the small t region and define the integrated fracture function as an integral over t up to a Q 2-dependent cut-off of order Q 2 , say e Q 2 , with e - 1 MAj , AX Ž x , z ,Q 2 . s
2
H0e Q dt M
j X A, A
Ž x , z ,t ,Q 2 . .
Ž 10 .
By taking the logarithmic derivative of Eq. Ž10. we have that E Q2 M j X Ž x , z ,Q 2 . E Q2 A, A E e Q2 s dt Q 2 M j X Ž x , z ,t ,Q 2 . E Q2 A, A 0
H
q e Q 2 MAj , AX Ž x , z , e Q 2 ,Q 2 . s
aS Ž Q 2 . 2p
2
1
H0e Q dtH x
1yz
du u
=Pi j Ž u . MAi , AX Ž xru, z ,t ,Q 2 . Fig. 1. Extended fracture function in the perturbative region of t.
q e Q 2 MAj , AX Ž x , z , e Q 2 ,Q 2 .
G. Camici et al.r Physics Letters B 439 (1998) 382–388
s
aS Ž Q 2 . 2p
du
1
Hx
Pi j Ž u . MAi , AX Ž xru, z ,Q 2 .
u
1yz
Ž 11 .
We see appearing an inhomogeneous term in the evolution equation of MA,j AX Ž x, z,Q 2 . which arises from the Q 2 dependence of the upper integration limit. This inhomogeneous term depends on the value of MA,j AX Ž x, z,t,Q 2 . in the perturbative region of t where Eq. Ž6. holds. By using the boundary condition Ekj Ž x ,Q 2 ,Q 2 . s d kj d Ž 1 y x .
e Q 2 MAj , AX Ž x , z , e Q 2 ,Q 2 . s
2p =Pˆi jl
s
r
w
2
.
2p
s
Hx
x
Dl , AX
aS Ž Q 2 . 2p
wyr
x
Hx
=Pˆi jl Ž u . D l , AX
z wyx
du
xqz
Ž w,Q .
,Q 2
zu x Ž 1 y u.
2
.
/
u
FAi Ž xru,Q 2 .
,Q 2 .
E EQ
s
2
1yz
m
MAj , AX Ž x , z ,Q 2 .
2p
1
Hx 1yz
X
dxx n MAj , AX Ž x , z ,Q 2 . s Mmj,nA A Ž Q 2 . .
Ž 15 . One can verify that Eq. Ž6. becomes aS Ž t . k l i i Mmj n Ž t ,Q 2 . s P F Ž t . Dmk Ž t . Enjl Ž t ,Q 2 . 2p t n m mqn Ž 16 .
Pmk nl i s
1
H0 du u
/
du u
=Pi j Ž u . MAi , AX Ž xru, z ,Q 2 .
m
n Ž 1 y u . Pˆil k Ž u .
Ž 17 .
Ž 13 .
1
n
j l
H0 du u E Ž u,t ,Q
2
..
Ž 18 .
The t evolution equation contains several terms: the first comes from the canonical scale dependence of the extended fracture function, the second from the scale dependence of a S , and the third from the scale dependence of the evolution kernel. There are then two inhomogeneous terms which follow from the t dependence of structure and fragmentation functions respectively. By explicitly deriving Eq. Ž16. with respect to t we get E j t M Ž t ,Q 2 . E t mn aS Ž t . j p s y d j p Ž1 q b0 aS Ž t . . q An 2p
ž
/
= Mmpn a S2
2
Ž t ,Q .
q
aS Ž Q 2 .
/
and
By substituting Eq. Ž13. in Eq. Ž11. the evolution equation for MA,j AX Ž x, z,Q 2 . appears to be Q2
1
H0 dzz H0
Enjl Ž t ,Q 2 . s
i A
u x Ž 1 y u.
ž
Hx
ž
2
/
Hzqx w Ž w y x . F Ž w,Q Dl , AX
FAi
,Q 2 d Ž xrr y 1 .
dw
1
ž / ž w
Hzqr w Ž w y r . z
du
F i Ž xru,Q 2 . x Ž 1 y u. A zu =Pˆi jl Ž u . D l , AX ,Q 2 , Ž 14 . x Ž 1 y u. that is exactly the one proposed in Ref. w1x. Furthermore the perturbative definition Ž6. can be used to derive the evolution with t of the extended fracture function. In order to make the notation less cumbersome, it is convenient to define q
where we have defined w6x
dw
1
r
ž / ž
aS Ž Q
=Pˆi jl
dr
1yz
xqz
Ž 12 .
on Eq. Ž6. we get for the inhomogeneous term, up to log e corrections,
aS Ž Q 2 .
x
aS Ž Q 2 . 2p
q e Q 2 MAj , AX Ž x , z , e Q 2 ,Q 2 . .
385
Ž t. 2
Ž 2p . t =Enjl
2
p p Pnkml i Aimqn Fmqn Ž t . Dmk Ž t .
Ž t ,Q . q
a S2 Ž t . 2 Ž 2p . t
= A kmp Dmp Ž t . Enjl Ž t ,Q 2 .
i Pnkml i Fmqn Ž t.
Ž 19 .
G. Camici et al.r Physics Letters B 439 (1998) 382–388
386
where A nji s
1
n
j
H0 du u P Ž u . .
Ž 20 .
i
We conclude this section by stressing that one can define the fracture function in a more general way as an integral up to an arbitrary Q 2-dependent integration limit MAj , AX Ž x , z ,Q 2 . s
2
H0t ŽQ .dt M
j X A, A
2
Ž x , z ,t ,Q 2 .
Ž 21 .
One could even define a more general fracture function introducing also a lower integration limit t 1Ž Q 2 . MAj , AX Ž x , z ,Q 2 . s
2
Ht tŽQŽQ. .dt M 2
2
j X A, A
Ž x , z ,t ,Q 2 . . Ž 24.
1
and the equation would have an additional inhomogeneous term taking into account the Q 2 dependence of the lower integration limit
E Q2
In this case the evolution equation reads
E Q2
Mmj n Ž Q 2 .
E Q2
MAj , AX Ž x , z ,Q 2 .
E Q2
s
s
aS Ž Q 2 . 2p
u
1yz
aS Ž t2 Ž Q
q
2
a S Ž t 2 Ž Q 2 . . Q 2 t 2X Ž Q 2 .
..
2
X
Q t2 Ž Q t2 Ž Q
2
2
.
.
du
rqz
r Ž 1 y u.
=Pˆik l Ž u . D l , AX
1yz
Hx
ž
r Ž 1 y u.
i Pnkml i Fmqn Ž t2 Ž Q 2 . .
dr r
y
FAi Ž rru,t 2 Ž Q 2 . . zu
t2 Ž Q 2 .
=Dmk Ž t 2 Ž Q 2 . . Enjl Ž t 2 Ž Q 2 . ,Q 2 .
a S Ž t 1 Ž Q 2 . . Q 2 t 1X Ž Q 2 . 2p
r
Hr
A nj p Mmp n Ž Q 2 .
2p
2p
=
2p
du
1
Hx
=Pi j Ž u . MAi , AX Ž xru, z ,Q 2 . q
aS Ž Q 2 .
,t 2 Ž Q 2 .
=Ekj Ž xrr ,t 2 Ž Q 2 . ,Q 2 .
t1 Ž Q 2 .
i Pnkml i Fmqn Ž t1 Ž Q 2 . .
=Dmk Ž t 1 Ž Q 2 . . Enjl Ž t 1 Ž Q 2 . ,Q 2 . .
/ Ž 22 .
Ž 25 .
Phenomenologically these two cases would correspond to the production of hadrons within the target fragmentation region with transverse momentum below t 2 Ž Q 2 . and between t 1Ž Q 2 . and t 2 Ž Q 2 . respectively.
or, by taking double moments,
E Q2
E Q2
s
Mmj n Ž Q 2 .
aS Ž Q 2 . 2p q
3. Sum rules
A nj p Mmp n Ž Q 2 .
a S Ž t 2 Ž Q 2 . . Q 2 t 2X Ž Q 2 . 2p
t2 Ž Q 2 .
In Ref. w1x it was shown that fracture functions obey the following momentum sum rule i Pnkml i Fmqn Ž t2 Ž Q 2 . .
Ý Hdz z MAj , A Ž x , z ,Q 2 . s Ž 1 y x . FAj Ž x ,Q 2 . Ž 26. X
X
A
=Dmk Ž t 2 Ž Q 2 . . Enjl Ž t 2 Ž Q 2 . ,Q 2 . .
Ž 23 .
In the case in which t 2 Ž Q 2 . s e Q 2 the evolution Eq. Ž14. is of course recovered.
which accounts for momentum conservation in the s-channel. In this section we want investigate if a momentum sum rule holds also in the t-channel.
G. Camici et al.r Physics Letters B 439 (1998) 382–388
Taking moments of Eq. Ž14. with m s 0, n s 1 and summing over j we get w9x
E Q2
EQ
s
X
2
Ý M01j, A A Ž Q 2 . j
aS Ž Q 2 . 2p
F1i , A Ž Q 2 .
X
1
H0 du u P
i
k
Ž 1 y u . D 0k , A Ž Q 2 . Ž 27 .
where we made use of the well-known property of the splitting function w6x
Ý A1ji s 0
Ž 28 .
j
and of the relation 1
H0 du Ž 1 y u . Ý Pˆ
jk
i
1
Ž u . s H du Ž 1 y u . Pi j Ž u . . 0
k
Ž 29 . Ž27. it appears that the derivative of From Eq. X j, A A Ž 2 . Ý j M01 Q receives contribution from the inhomogeneous term in the evolution equation. This fact suggests that the sum rule could be violated because at values of t of order Q 2 the current contribution becomes important and the two production mechanisms cannot be disentangled. As far as the extended fracture function is concerned, since MA,j AX Ž x, z,t,Q 2 . obeys the DGLAP evolution equation, it follows that
E Q2
EQ
1yz
2
ÝH j
0
x dx MAj , AX Ž x , z ,t ,Q 2 . s 0
Ž 30 .
and momentum conservation in the t channel is recovered. In the perturbative region of t, by using Eq. Ž6., we find X
Ý M01j, A A Ž t ,Q 2 . j
s
aS Ž t . 2p t
F1i , A Ž t .
X
1
H0 du uP
i
k
Ž 1 y u . D 0k , A Ž t . . Ž 31 .
4. Summary In this note we showed that a formulation of perturbative processes in the target fragmentation
387
region can be given by using the formalism of Jet Calculus w6x. A perturbative evaluation in terms of evolution equation and anomalous dimensions of extended fracture function is consistent with the evolution equations proposed in Ref. w1x and w2x for ordinary and extended fracture functions respectively. We showed that the inhomogeneous term in Eq. Ž14. precisely stems from the integration over momentum transfer t. Moreover in this formalism the extension to nextto-leading order seems quite natural and suggests that Eqs. Ž9. and Ž14. keep the same structure in the two-loops approximation, analogously to what happens for ordinary structure and fragmentation functions w10x. In the region L2QC D < t < Q 2 the hard semi-inclusive cross section develops large log Q 2rt corrections which need to be resummed. Our approach is based on the idea of absorbing such corrections in the definition of MA,i AX Ž x, z,t,Q 2 ., namely, logQ 2rt are fully contained in the evolution kernel Eik Ž x,t,Q 2 .. These logs, which can be treated with standard renormalization group techniques, are new and potentially useful to understand the dynamics of hadro-production in the target fragmentation region. These logs could affect significantly observables like multiplicities and particle distributions and we believe that a phenomenological study of the role of these corrections would be useful. As originally proposed in Ref. w1x and in Ref. w11x, there now seems to be a widespread consensus that the Q 2 evolution in the diffractive regime can be described in terms of the perturbative QCD formalism w4,12,13x. The data taken by H1 collaboration w14x, showing an evidence of a consistent logarithmic scaling violation in diffractive channels have further stressed that such an interpretation is not far from the experimental observation. All this is consistent with a description in terms of fracture functions. Our results agree with such an expectation, although in the limit z ™ 1 our formulae will show the appearance of large logŽ1 y z . factors which should be resummed as well. For the treatment of such a singular region we believe an approach can be used close to the one followed in the inclusive case. As a concluding remark we want to stress that the results presented here completely agree with those of
388
G. Camici et al.r Physics Letters B 439 (1998) 382–388
Ref. w2x, obtained within a quite different framework, namely, that of a generalized cut vertex expansion which, as a generalization of Operator Product Expansion, appears to be founded on a firmer theoretical standpoint. Once more the equivalence of parton-like and OPE-like approaches is verified.
w3x w4x w5x w6x w7x w8x w9x
Acknowledgements Many of the arguments covered in this letter originate from discussions with G. Veneziano. We also thank G. Veneziano for useful comments, and S. Catani, D. Graudenz and J. Kodaira for conversations. One of us ŽM.G.. would like to thank the Department of Physics of the university of Florence for the hospitality at various stages of this work.
w10x w11x w12x
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w14x
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