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A modified parton model for reactions with time-like photons
Volume 56B, number 4
PHYSICS LETTERS
12 May 1975
A MODIFIED PARTON M O D E L F O R R E A C T I O N S W I T H T I M E - L I K E PHOTONS J. LAYSSAC and F.M. RENARD Ddpartement de Physique Mathdmatique*, Universitd des Sciences et Techniques du Languedoc, 34060 Montpellier Cedex, France
Received 20 March 1975 A time-like photon is considered as a kind of mesonic state with valence + sea multipartons configuxations;consequences for e+e- annihilation and leptons production in ha&ohio collisionsare examined.
It is well-known [1 ] that the usual quark parton model fails to explain the behaviour of the e+e - annihilation (Oret and inclusive distributions) as well as the leptons production in hadronic collisions. On the other hand, for space-like photons, the deep inelastic scattering of leptons on hadrons is remarkably well described by this model; it reveals however a very special content of the nucleon [2] : non strange quarks 45%, strange quarks and antiquarks 6%, gluons 49%, the contribution of the q?:lpairs of the sea being localized for fractions of the nucleon momentum x < 0.4. In this paper we assume that the failure of the usual phenomenological parton models in reactions where a parton-antiparton pair is produced comes from the ignorance of sea and gluons effects. This is already obvious at low q2 where the occurrence of the vector mesons is a particular case of interactions between the quarks. We propose the simple phenomenological model: a time-like photon is a kind of mesonic state made of a pair of valence quarks plus a sea of other pairs and gluons. In e+e - annihilation into hadrons, the photon decays into its real constituents (which depend upon its mass x/rq2-); in h + h' -* £+ + £- + X the photon state is constructed by taking in h and in h' the partons, pairs and gluons, required by its valence + sea configurations. In the deep inelastic scattering of leptons on hadrons, the photon is space-like (out of the mass-shell of any meson) and is there only for measuring for the leptons the charges of the partons. In such a model there is no simple connection between the spacelike and time-like regions a situation which is theoretically allowed [3] (due to the presence of singularities for time-like q2 in the structure functions). Let us write the time-like photon state, like any other hadronic state, as a superposition o f n partons configurations with the probability P(n). The maximum number nmax and perhaps also these probabilities may depend upon the mass q2 of the photon (i.e. P ( n , q2)), but in any case the normalization condition ~ nmax P(n, q2) = 1 ensures that the asymptotic behaviour of the total cross section of e+e - annihilation is not modified (see fig. 1) 4~ro~___~. 2 Oret = 3q 2 For leptons pairs production in hadronic collisions, the Drell-Yan meclaanism [4] is now modified following fig. 2 and for example the mass-distribution is given by: do _ 41tel2 dq 2 3q 4
2
~,N
fdx I . . . d x N d x
I ...dX'N, 5 ( ( Z N x ) ( y , N ' x ' ) - 7")
(2)
X (Y.,Nx)(Y-,"V'x')fha(XI,X2, •.. X N )...eh'. , ). f ~ I.X,1 , X 2, , ... XN,
The functions f~a(x 1 .... X N ) give the probabilities of finding N partons (including a) inside the hadron h with me* Physique Math~nlatiaue et Th~orique, Equipe de Recherche Associ~eau C.N.R.S. 364
Volume 56B, number 4
PHYSICS LETTERS
Fig. 1. Diagramfor e+e" annihilation into hadxons.
12 May 1975
Fig. 2. Diagramfor leptons pairs production in hadxonic collisions.
menta xiP h (i = 1,2 ..... N); these probabilities generalize the one fha(X) which appears usually in DIS for a single parton a; r = q2/s. For N = N ' - 1 and P(2) = 1, eq. (2) matches the Drell-Yan Formula [4]. The behaviour of these multi-partons distributions is easy to get in the Bjorken-Paschos model [5] with a uniform repartition of the fractions x F [[~y m-'~N-1 } f(xl, "" ,XN) = (--1) N-1 __--ZT-;~• f(Y) (3)
y=~Nx
it is then a function of ZNx solely, where f(x) is still the probability for Finding one parton with the fraction x and which is available from fits of DIS of leptons on hadrons. Let us consider the simple description where the hadron h is made of a valence configuration plus a sea of pairs and gluons distributed with a Poisson shape ph(m) = e-gg m/m!.
(4)
The normalizations of the multi-partons distributions are then
fdxl ...
ra
....
~X N )
= (m(m
f dx~(x)
-
1 ) ( m - N + 1 )) = g N - 1 .
(5)
(m)
Using f dx I ... dx N g(~-,Nx) = f dy D,N-1/(N - 1)!] g(y), a simple approximation consistent with eq. (3) and (5) is:
fha(Xl,... ,XN) -
(N--l)!
(~,Nx)N-1
gN_l fha(ZNx)"
(6)
In this case one gets: do _ 4~'a 2 dr/2 3q 4 ~a ~2 N,N~'P(N+N')gN+N'-2fdydy'8(YY'-r)yy'J~aa(Y)f~(Y')
(7)
and the correction factor with respect to the Drell-Yan formula is:
K = ~ P(N+N')g N+N'-2. N,N'
(8)
If the probability for partons configurations inside the sea of the photon is also Poissonian:
P(n) = e-gg'n/n!
(9)
then K = (1 +gg') exp [ g ' ( g - 1)].
(10) 365
Volume 56B, number 4
PHYSICS LETTERS
12 May 1975
g a n d g ' mean roughly the average numbers of pairs and gluons inside the hadrons and inside the photon mesonic state. They control the enhancing or depressing character of K; for example if g = g ' : K = 1 for g = 0, Kmi n ~ 0.17 for g ~ 0.25 and K > 1 for g > 0.5. In principle a good knowledge of the DIS on hadrons h would determine g. There are good indications [2] for a weak sea contribution localized at small x (and even negligible for x > 0.4). This would mean that a more realistic model would give in eq. (5), (8), (10) g(r) instead of constant g, with for example: g(r) --g(1 - z) k.
(11)
In this case the shape of the mass distribution do/dq 2 and o f the single lepton transverse momentum distribution / ° d o / d 3 l, could be modified: for r -~ 0 or l T -~ 0, g would be large and K > 1
for r -~ 1 or l T large, g would be small and K ~< 1.
This is exactly what seems to be required by experiments [6] ; for l T distributions the usual parton models [7] give results too small for small l T but which flatten for large i T ; for q2 distributions the experimental situation is not clear but it seems that the cross section drops very rapidly down for large q2 and perhaps more rapidly than the simple Drell-Yan term. The second coupling g' could be reached in e+e - annihilation into hadrons. With this picture the final hadrons are connected to the multipartons configurations and roughly:
g' ~ (n(n - 1))/(n) measurable for example among pions. The inclusive distribution q2 do/dx in e+e - -+ zr + X (x = 2ETr/x/~) is sensitive to the opening o f multipartons configurations through P(n) or P(n, q2). It is tempting to relate the violation of scaling observed [1 ] in q2 do/dx for x < ½ and 3.0 ~< x / ~ ~< 4.8 GeV with the existence of a non negligible sea distribution inside the ha&on for the same range of x. It would be like if by increasing q2 one reveals progressively the more and more inner structure of the hadronic state which is necessarily virtual in DIS; of course in this energy range o f e+e - annihilation there are also threshold effects (for ex o f charmed particles) which produce natural scaling violations observables also in o~ot. But it would be interesting to verify if for higher q2, Otot having the 1/q2 scaling behaviour, the inclusive distribution q2 do/dx will scale and allow the determination o f P(n) and g ' for a new kind of connection between DIS, e+e - and leptons pairs production.
References [1 ] B. Richter, Ptoc. XVII Int. Conf. on High energy physics, London (1974) IV-37; P.V. Landshoff, Proc. XVII Int. Conf. on High energy physics, London (1974) V-57. [2] D.C. Cundy, Proc. XVII Int. Conf. on High energy physics, London (1974) IV-131. P.V. Landshoff, Ikoc. Cern School of Phys. (1974) Cern 74-22; G. AltareUi, Proc. Gff-sur-Yvette Summer School, IN2P3 (1974) 161. [3] R. Gatto and G. Preparata, Nucl. Phys. B47 (1972) 313. [4] S.D. Drell and T.M. Yan, Ann. of Phys. 66 (1971) 578. [5] J.D. Bjorken and E. Paschos, Phys. Rev. 185 (1969) 1975. [6] LM. Lederman, Proc. XVII Int. Conf. on High energy physics, London (1974) V-55; J. Ckristenson et al., Phys. Rev. Lett. 25 (1970) 1523; J.J. Aubert et al., Phys. Rev. Lett. 33 (1974) 1404. [7] F.M. Renard, preprint Montpellier PM]75/3.
366
PHYSICS LETTERS
12 May 1975
A MODIFIED PARTON M O D E L F O R R E A C T I O N S W I T H T I M E - L I K E PHOTONS J. LAYSSAC and F.M. RENARD Ddpartement de Physique Mathdmatique*, Universitd des Sciences et Techniques du Languedoc, 34060 Montpellier Cedex, France
Received 20 March 1975 A time-like photon is considered as a kind of mesonic state with valence + sea multipartons configuxations;consequences for e+e- annihilation and leptons production in ha&ohio collisionsare examined.
It is well-known [1 ] that the usual quark parton model fails to explain the behaviour of the e+e - annihilation (Oret and inclusive distributions) as well as the leptons production in hadronic collisions. On the other hand, for space-like photons, the deep inelastic scattering of leptons on hadrons is remarkably well described by this model; it reveals however a very special content of the nucleon [2] : non strange quarks 45%, strange quarks and antiquarks 6%, gluons 49%, the contribution of the q?:lpairs of the sea being localized for fractions of the nucleon momentum x < 0.4. In this paper we assume that the failure of the usual phenomenological parton models in reactions where a parton-antiparton pair is produced comes from the ignorance of sea and gluons effects. This is already obvious at low q2 where the occurrence of the vector mesons is a particular case of interactions between the quarks. We propose the simple phenomenological model: a time-like photon is a kind of mesonic state made of a pair of valence quarks plus a sea of other pairs and gluons. In e+e - annihilation into hadrons, the photon decays into its real constituents (which depend upon its mass x/rq2-); in h + h' -* £+ + £- + X the photon state is constructed by taking in h and in h' the partons, pairs and gluons, required by its valence + sea configurations. In the deep inelastic scattering of leptons on hadrons, the photon is space-like (out of the mass-shell of any meson) and is there only for measuring for the leptons the charges of the partons. In such a model there is no simple connection between the spacelike and time-like regions a situation which is theoretically allowed [3] (due to the presence of singularities for time-like q2 in the structure functions). Let us write the time-like photon state, like any other hadronic state, as a superposition o f n partons configurations with the probability P(n). The maximum number nmax and perhaps also these probabilities may depend upon the mass q2 of the photon (i.e. P ( n , q2)), but in any case the normalization condition ~ nmax P(n, q2) = 1 ensures that the asymptotic behaviour of the total cross section of e+e - annihilation is not modified (see fig. 1) 4~ro~___~. 2 Oret = 3q 2 For leptons pairs production in hadronic collisions, the Drell-Yan meclaanism [4] is now modified following fig. 2 and for example the mass-distribution is given by: do _ 41tel2 dq 2 3q 4
2
~,N
fdx I . . . d x N d x
I ...dX'N, 5 ( ( Z N x ) ( y , N ' x ' ) - 7")
(2)
X (Y.,Nx)(Y-,"V'x')fha(XI,X2, •.. X N )...eh'. , ). f ~ I.X,1 , X 2, , ... XN,
The functions f~a(x 1 .... X N ) give the probabilities of finding N partons (including a) inside the hadron h with me* Physique Math~nlatiaue et Th~orique, Equipe de Recherche Associ~eau C.N.R.S. 364
Volume 56B, number 4
PHYSICS LETTERS
Fig. 1. Diagramfor e+e" annihilation into hadxons.
12 May 1975
Fig. 2. Diagramfor leptons pairs production in hadxonic collisions.
menta xiP h (i = 1,2 ..... N); these probabilities generalize the one fha(X) which appears usually in DIS for a single parton a; r = q2/s. For N = N ' - 1 and P(2) = 1, eq. (2) matches the Drell-Yan Formula [4]. The behaviour of these multi-partons distributions is easy to get in the Bjorken-Paschos model [5] with a uniform repartition of the fractions x F [[~y m-'~N-1 } f(xl, "" ,XN) = (--1) N-1 __--ZT-;~• f(Y) (3)
y=~Nx
it is then a function of ZNx solely, where f(x) is still the probability for Finding one parton with the fraction x and which is available from fits of DIS of leptons on hadrons. Let us consider the simple description where the hadron h is made of a valence configuration plus a sea of pairs and gluons distributed with a Poisson shape ph(m) = e-gg m/m!.
(4)
The normalizations of the multi-partons distributions are then
fdxl ...
ra
....
~X N )
= (m(m
f dx~(x)
-
1 ) ( m - N + 1 )) = g N - 1 .
(5)
(m)
Using f dx I ... dx N g(~-,Nx) = f dy D,N-1/(N - 1)!] g(y), a simple approximation consistent with eq. (3) and (5) is:
fha(Xl,... ,XN) -
(N--l)!
(~,Nx)N-1
gN_l fha(ZNx)"
(6)
In this case one gets: do _ 4~'a 2 dr/2 3q 4 ~a ~2 N,N~'P(N+N')gN+N'-2fdydy'8(YY'-r)yy'J~aa(Y)f~(Y')
(7)
and the correction factor with respect to the Drell-Yan formula is:
K = ~ P(N+N')g N+N'-2. N,N'
(8)
If the probability for partons configurations inside the sea of the photon is also Poissonian:
P(n) = e-gg'n/n!
(9)
then K = (1 +gg') exp [ g ' ( g - 1)].
(10) 365
Volume 56B, number 4
PHYSICS LETTERS
12 May 1975
g a n d g ' mean roughly the average numbers of pairs and gluons inside the hadrons and inside the photon mesonic state. They control the enhancing or depressing character of K; for example if g = g ' : K = 1 for g = 0, Kmi n ~ 0.17 for g ~ 0.25 and K > 1 for g > 0.5. In principle a good knowledge of the DIS on hadrons h would determine g. There are good indications [2] for a weak sea contribution localized at small x (and even negligible for x > 0.4). This would mean that a more realistic model would give in eq. (5), (8), (10) g(r) instead of constant g, with for example: g(r) --g(1 - z) k.
(11)
In this case the shape of the mass distribution do/dq 2 and o f the single lepton transverse momentum distribution / ° d o / d 3 l, could be modified: for r -~ 0 or l T -~ 0, g would be large and K > 1
for r -~ 1 or l T large, g would be small and K ~< 1.
This is exactly what seems to be required by experiments [6] ; for l T distributions the usual parton models [7] give results too small for small l T but which flatten for large i T ; for q2 distributions the experimental situation is not clear but it seems that the cross section drops very rapidly down for large q2 and perhaps more rapidly than the simple Drell-Yan term. The second coupling g' could be reached in e+e - annihilation into hadrons. With this picture the final hadrons are connected to the multipartons configurations and roughly:
g' ~ (n(n - 1))/(n) measurable for example among pions. The inclusive distribution q2 do/dx in e+e - -+ zr + X (x = 2ETr/x/~) is sensitive to the opening o f multipartons configurations through P(n) or P(n, q2). It is tempting to relate the violation of scaling observed [1 ] in q2 do/dx for x < ½ and 3.0 ~< x / ~ ~< 4.8 GeV with the existence of a non negligible sea distribution inside the ha&on for the same range of x. It would be like if by increasing q2 one reveals progressively the more and more inner structure of the hadronic state which is necessarily virtual in DIS; of course in this energy range o f e+e - annihilation there are also threshold effects (for ex o f charmed particles) which produce natural scaling violations observables also in o~ot. But it would be interesting to verify if for higher q2, Otot having the 1/q2 scaling behaviour, the inclusive distribution q2 do/dx will scale and allow the determination o f P(n) and g ' for a new kind of connection between DIS, e+e - and leptons pairs production.
References [1 ] B. Richter, Ptoc. XVII Int. Conf. on High energy physics, London (1974) IV-37; P.V. Landshoff, Proc. XVII Int. Conf. on High energy physics, London (1974) V-57. [2] D.C. Cundy, Proc. XVII Int. Conf. on High energy physics, London (1974) IV-131. P.V. Landshoff, Ikoc. Cern School of Phys. (1974) Cern 74-22; G. AltareUi, Proc. Gff-sur-Yvette Summer School, IN2P3 (1974) 161. [3] R. Gatto and G. Preparata, Nucl. Phys. B47 (1972) 313. [4] S.D. Drell and T.M. Yan, Ann. of Phys. 66 (1971) 578. [5] J.D. Bjorken and E. Paschos, Phys. Rev. 185 (1969) 1975. [6] LM. Lederman, Proc. XVII Int. Conf. on High energy physics, London (1974) V-55; J. Ckristenson et al., Phys. Rev. Lett. 25 (1970) 1523; J.J. Aubert et al., Phys. Rev. Lett. 33 (1974) 1404. [7] F.M. Renard, preprint Montpellier PM]75/3.
366