- Email: [email protected]
Exclusive two ϱ0-mesons production in γγ∗ collision
as.H---b!EiJ
SUPPLEMENTS
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 126 (2004) 277-282
Exclusive two pO-mesons production I.V. Anikin”,
in yy* collision
B. Pireb and 0. V. TeryaevC
a Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia CPhT, Ecole Polytechnique, F-91128 Palaiseau, France I b CPhT, Ecole Polytechnique, F-91128 Palaiseau, France c Bogoliubov
Laboratory
of Theoretical
Physics, JINR, 141980 Dubna, Russia
We give a theoretical description of the L3 Collaboration data on the cross-section of the exclusive two p-mesons production in yy* collision with large photon virtuality. These data prove the scaling behaviour of the leading twist generalized distribution amplitude of vector mesons. They thus prove the relevance of a partonic description of the exclusive process yy* + pop0 at Q2 >_ 1.2 GeV2 and W <_ 3.0 GeV.
1. Introduction Two-photon collisions provide a tool to study a variety of fundamental aspects of QCD and have long been a subject of great interest (cf, e.g., [ 1, 2, 31 and references therein). A peculiar facet of this interesting domain is exclusive two hadron production in the region where one initial photon is highly virtual (its virtuality being denoted as -Q2) but the overall energy (or invariant mass of the two hadrons) is small [ 41. This process factorizes [ 5, 61 into a perturbatively calculable, short-distance dominated scattering y*y --+ qtj or y*y -+ gg, and nonperturbative matrix elements measuring the transitions qtj -+ AB and gg + AB. These matrix elements have been called generalized distribution amplitudes (GDAs) to emphasize their close connection to the distribution amplitudes introduced many years ago in the QCD description of exclusive hard processes [ 71. In this paper, we focus on the process yy* + pop0 which has recently been observed at LEP in the right kinematical domain [ 81. We calculate the differential cross-section of the considered processes as a function of Q2 and compare it to the experimental data.
2.
Kinematics The reaction which we study here is:
e(k) + 4)
-+ 4k’) + 40
+ pa(n)
+
p”(P2)
(1)
where the initial electron e(lc) radiates a hard virtual photon with momentum q = k - k’ ; in other words, the square of virtual photon momentum q2 = -Q2 is very large. This means that the scattered electron e(k’) is tagged. To describe reaction (l), it is useful to consider, at the same time, the sub-process of reaction (1) : e(k) + y(d)
-+ 4k’) + Pi
+ P~CPZ).
(2)
Regarding the other photon momentum qf = l-l’, we assume that, firstly, its momentum is collinear to the electron momentum I and, secondly, that equal to zero, which is a 4‘2 is approximately usual approximation when the second lepton is untagged. Let us now pass to a short discussion of kinematics in the yy* center of mass system. We adopt the z axis directed along the threedimensional vector q, and the p-meson momenta are in the (z, z)-plane. So, we write for momenta in the c.m. system :
9 = (90,0, 0, 4, m = (18, plsino, 0920-5632/s - see front matter 0 2004 Published by Elsevier B.V. doi: 10.1016/~j.nuc1p1~ysbps.2003.1 I .047
0, PICOS~)
(3)
I. R Anikin et ul./Nuclear
278
Physics B (Pmt. Suppl.) 126 (2004) 277-282
Also, we need to write down the Mandelstam S-variables for the electron-positron (1) and electron-photon (2) collisions:
Within this frame the transverse component of the transfer momentum AT = (0, AT, 0) is given by
se, = (k + E)2,
*$=--A$=
s,, = (k + q’)2.
(4
Neglecting the lepton masses, these variables can be rewritten as S ee M 2(k. I),
se, M 2(k. q’) = 22See,
(5)
where the fraction 22 defined as qh = x210 is introduced (see, for instance [ 91).
(W2--4naz)sin?8.
We now come to the parameterization of the relevant matrix elements. Keeping the terms of leading twist 2, the vector and axial correlators can be written as (see, for instance, [ lo], where the crossed case of the generalized parton distributions in a spin-l particles is considered): (Pl,Xl;P2,X2113;(0)~,~(Xn)lO)
3. Parameterization of p-matrix and their properties
(p . n) = 1.
(6)
In this basis, the pmesons can be written as
momenta
1-
(I
1+
As usual, we introduce hadronic momenta: -p;,
pl and pz
A; 2 ’ 3 2
(7)
the sum and difference of
P’” =pg
The skewedness parameter
+p’i”.
(8)
t is defined by
A en = -eP.n,
(9)
or
I+(-
lfPCOS8 2
2
In (13) and (14), the tensor structures V$(pl,p~, n) and Af$(p,,p,, n) depend on the vectors pl, p2 and n. Notice that due to the parity invariance the tensors VA2 may be written in terms of five tensor structures while the tensors A$ are linear combinations of four independent structures.
1-g. d-----
=c.
g,T,W/)V(y, case, W2)+
(10)
Notice that the relation between our definition and definition in, for instance, [ 91 reads
T
of yy* 4 pop0 subprocess
In this section, we pass to the consideration of y(q’)y*(q) + p”(pl)po(pz) subprocess. Following [ 111, the amplitude of this subprocess including the leading twist-2 terms can be written as
T,v= ,p=
(13)
Here Xi and X2 are the helicities of spin-l hadrons and 2 denotes the Fourier transformation with measure (21 = Xn,zs = 0) [ 111: $p(y) = dy e--iY Wl-ti(l-Y) PZZ (15)
4. Amplitude Ap =p;
2
elements
Let us first introduce the basis light-cone vectors. We adopt a light-cone basis consisting of two light-like vectors p and n of mass dimension 1 and -1, respectively. In other words, they obey the following conditions : p2 = n2 = 0,
(12)
@+(Y)A(Y,
~0~8, W”> 1 ,
(16)
where (17)
i. R Anikin et al. /Nuclear Physics B (Proc. Suppl.) 126 (2004) 277-282
In (16), the scalar and pseudo-scalar functions (V, A) denote the following contraction V(y,cosB,W2)
A(y,
cos 8, W”)
12
=
-Q~czz
= (1 - a2)E,2sin2 omaz.
=
zz .;(i)$T’4v.
du(e+ty -+ e+pOpO) _ a3 ,!3 1 dW2 dcos6’d# 16~ S& 0”
1- 25'e,(Q2+ W2-se,) (Q2 + W2)2
(19)
Here, in the yy* c.m. frame system, the photon polarization vectors read
]A(+,+) (~0s 0, W”) I2
4 jV(cos0, (
(20)
e+e- + e+e-pop’)
JJ
=
IA(cos
W2)j2).
(25)
summed
W2)12 = P=-(pl)Pp1pz(p2)
(26)
dy&(yl)HfPT
v(yl,cosh’,
W2)
dy&(yz)HjPP’
V(y2rcos8, W”)
where the Weizsacker-Williams defined as usual : Fww(xz)
=
I+
(I2x2
-- 1 - x2 ’
~2)~~~
= P”‘““(p1)Pp1p2(pz)
dyz~+(yz)H~'"(~z,cose,
du(e+y -+ e+pOpO) dW2 dcos8 d4 ’
3vw(x2)
8, W2)12
function
(21)
(27)
W”) yc~ Jdyl~+(yl)H~P’A(yl,co
dW2 dcos8 d$ dxz
x2
W2)j2 -I- ]A(cosO,
cross-sections
We will now concentrate on the calculation of the differential cross-section of (1). Using the equivalent photon approximation we find the expression for the corresponding cross-section :
..
=
In (25), the squared and polarization functions IV12 and IAl2 read : lV(cos8,
for the real and virtual photons, respectively.
(24)
where [A(+,+)(cosW”)~~
5. Differential
(23)
The angle Q,,, in (23) is determined by the acceptance of a lepton in the detector (see, for instance, [ 91) and the value of & is 91 GeV at LEPl and 195 GeV at LEP2. In (21), the cross-section for the subprocess can be calculated directly ; we have
The next objects of our consideration are the helicity amplitudes that are obtained from the usual amplitudes after multiplying by the photon polarization vectors &)
and the value Q’ 2 is defined as Q maz
=
279
W”),
where
FWW is The helicity amplitude after the integration cosB is implemented may be written as
Q’2(x2)
over
m2
(22)
F(+,+)(W2) =J dcose ~-$+:+~b~4W")/~. (29)
280
I. r Anikin et al. /Nucleur Physics B (Prvc. Suppl.) 126 (2004) 277-282
So, the cross-section takes the form e+e- -+ e+e-pop’) a4 1 dxzFww(xd 167~~
Finally, the cross-section (30) is expressed through these three phenomenological parameters in the following simple way:
=
-j$(e+e(
1 16x;E,4Q2 I dW2P F(+,+) (W”>-
Fww
-
2
(30)
’
Notice that the integrated amplitude F(+,+) is independent of Q2 up to logarithms. This is quite natural for the cases where the factorization theorem is applied. Besides, the exact W2 dependence of this amplitude remain unknown unless some modeling is used. However, the mean value theorem gives the possibility to reduce the three different integrals over W2 in (30) to one integration. Indeed, for our case the mean value theorem reads :
I
,jW2 pF(+>+) cw2)= Q2+W2 1
Q2 + (WI)” I
I
dW2P F(,,,)
W”>>
dW2 p F(+,+) cw2) _ (Q2 + W2)2 -
dW2P F(+,+) W”) (31) (Q2 + &2j212 I with two phenomenological parameters (WI) , (W2). Because of that the values of these parameters, generally speaking, have the same order of magnitude as Q, i.e. are not much less than Q, we need to keep (I&‘)-parameters in the prefactors of (31). Thus, we can see from (31) that it is useful to introduce a third phenomenogical parameter as W?L Cl =
I YL
dW2P F(+,+)(W2L
= &
Cl
I
dx2
1 2 16x;E,4Q2 + Q2(Q2 + W2)2)2 1
1 dW2 @F(+,+) cw2) + 2x2 E;Q2 I Q2+W2 dW2 P F(,,,) W”) (Q2 + W2)2 &2 I
(~2)
+ e+e-pop’)
(32)
where the integration over W2 runs into the limits dictated by the experiment.
2x&Q2(Q2 + (WI)“)
’
(33)
Within the analysis implemented by the L3 Collaboration, the value W is in the interval 1.1 GeV < W < 3.0GeV. Hence we are able to conclude that the phenomenological parameters (WI) and (W2) may take, generally speaking, any values inside this interval. 6. Numerical
results
and Discussion
In the previous section we obtained a simple expression for the cross-section as a function of three parameters which are (WI), (WZ) and Cr. Let us now make a fit of these phenomenological parameters in order to get the best description of experimental data. The best values of the parameters can be found by the method of least squares, x2-method, which flows from the maximum likelihood theorem (see, for instance, [ 121). As usual, the x2-sum as a function of parameters is written in the form : (34) where P = {(WI), (WZ), Cr} denotes the set of fitted parameters ; ozZp and cf” are the experimental measurements of the cross-section and its theoretical estimations ; Sai are the statistical errors. The experimental data for the cross-section of the exclusive double p” production were taken from the measurement of the L3 collaboration at LEP [ 81. Further, minimizing x2-sum in (34) with respect to the parameters P we find that the set of solutions Pmin with the confidence intervals are the following : Cl = 1.0 f 0.12 GeV2, (W2) = 1.2 f 0.08 GeV, (WI) = 2.9 f 1.8 GeV. (35)
I. R Anikin
With this the magnitude and, therefore, we have
et al. /Ntrclear
of x2 is equal to
Physics
.25
X2 - 0.25 < 1 degree of freedom The confidence intervals were defined for the case of one-standard deviation. The graphics with the experimental data and theoretical cross-section for the fitted parameters (35) are drawn on Fig. 1. The obtained value of the normalization constant Ci of order one is probably related to the energy momentum conservation, which should define the scale of GDA, like in two pions [ 131 case. Besides, notice that within L3 experiment the interval of changing for W in the yy* center of mass is fixed as 1.1 GeV < W < 3.0 GeV. Therefore, we are able to note, taking into account (35), that the confidence intervals for parameters (WI) lie into the whole available interval for W. In other words, we can infer that the theoretical crosssection depends on (WI) slightly. Indeed, despite all the three terms in (33) are participating in our analysis, the behaviour of the cross-section is mainly dictated by the third term in (33).
1
B (Proc.
Suppl.)
126 (2004) 277-282
281
7. Conclusion Data thus are in full agreement with the theoretical expectation on the Q2 behaviour of the cross section. Because of that the effective structure function F(+,+j is independent of Q2 up to the logarithms, as it was said before, our analysis of the data is a direct proof of the relevance of the partonic description of the process yy* -+ pop0 in the kinematics of the L3 experiment. Much more can be done if detailed experimental results are collected. For instance the angular dependence of the final state is a good test of the validity of the asymptotic form of the generalized distribution amplitudes. The spin structure of the final state, if elucidated, would allow to disentangle the roles of the nine generalized distribution amplitudes. The W2 behaviour of the cross section may have some interesting features. It depends much on the possible resonances which are able to couple to two p mesons. Its Fourier transform allows to have access to an impact picture of exclusive fragmentation [ 141. The p+pchannel may be calculated along the same lines. In that case a brehmstrahlung subprocess where the mesons are radiated from the lepton line must be added. The charge and spin asymmetries then come from the interference of the two processes. In that way one may also study the large contribution of 47r continuum. These items will be discussed in a forthcoming publication.
0.1 0.01 0.001
We acknowledge useful discussions and correspondance with M. Diehl, A. Nesterenko and I. Vorobiev. REFERENCES
Figure 1. Cross-section do/dQ2[pb/GeV2] as a function of Q2 . The theoretical cross-section is plotted for the best fitted parameters which are Cl = 1 .OGeV2 , (WI) = 2.9GeV and (WZ) = 1.2 GeV. Experimental data [ 81 come from LEPl and LEP2 runs.
1. H. Terazawa, Rev. Mod. Phys. 45 (1973) 615. 2. V.M. Budnev et al., Phys. Rept. 156 (1975) 181. 3. S.J. Brodsky, hep-ph/9708345, talk presented at PHOTON 97, Egmond aan Zee, Netherlands, May 1997; M.R. Pennington, Nucl. Phys. B (Proc. Suppl.) 82 (2000) 291 [hep-ph/9907353]. 4. M. Diehl, T. Gousset, B. Pire and O.V. Teryaev, Phys. Rev. Lett. 81 (1998) 1782
282
9. 10. 11. 12.
13. 14.
I. R Anikin et ul. /Nucleur Physics B (Pvoc. Suppl.) 126 (2004) 277-282
[hep-ph/9805380]; M. Diehl, T. Gousset and B. Pire, Nucl. Phys. B (Proc. Suppl.) 82 (2000) 322 [hepph/9907453]. D. Miiller et al., Fortschr. Phys. 42 (1994) 101 [hep-ph/9812448]. A. Freund, Phys. Rev. D61 (2000) 074010 [hep-ph/9903489]. G.P. Lepage and S.J. Brodsky, Phys. Rev. D22 (1980) 2157. L3 Collaboration, talk presented by S. Nesterov in “PHOTON 03” Workshop, April, 711 2003, Frascati, Italy (these proceedings) M. Diehl, T. Gousset and B. Pire, Phys. Rev. D 62 (2000) 073014 E. R. Berger, F. Cano, M. Diehl and B. Pire, Phys. Rev. Lett. 87 (2001) 142302 I. V. Anikin and 0. V. Teryaev, Phys. Lett. B 509 (2001) 95. S. L. Meyer, “Data analysis for scientists and engineers 1), Wiley series in probability and mathematical statistics, Edt. R. A. Bradley and J. S. Hunter, (1975) M. V. Polyakov, Nucl. Phys. B 555 (1999) 231 [hep-ph/9809483]. B. Pire and L. Szymanowski, Phys. Lett. B 556 (2003) 129 [hep-ph/0212296].
SUPPLEMENTS
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 126 (2004) 277-282
Exclusive two pO-mesons production I.V. Anikin”,
in yy* collision
B. Pireb and 0. V. TeryaevC
a Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia CPhT, Ecole Polytechnique, F-91128 Palaiseau, France I b CPhT, Ecole Polytechnique, F-91128 Palaiseau, France c Bogoliubov
Laboratory
of Theoretical
Physics, JINR, 141980 Dubna, Russia
We give a theoretical description of the L3 Collaboration data on the cross-section of the exclusive two p-mesons production in yy* collision with large photon virtuality. These data prove the scaling behaviour of the leading twist generalized distribution amplitude of vector mesons. They thus prove the relevance of a partonic description of the exclusive process yy* + pop0 at Q2 >_ 1.2 GeV2 and W <_ 3.0 GeV.
1. Introduction Two-photon collisions provide a tool to study a variety of fundamental aspects of QCD and have long been a subject of great interest (cf, e.g., [ 1, 2, 31 and references therein). A peculiar facet of this interesting domain is exclusive two hadron production in the region where one initial photon is highly virtual (its virtuality being denoted as -Q2) but the overall energy (or invariant mass of the two hadrons) is small [ 41. This process factorizes [ 5, 61 into a perturbatively calculable, short-distance dominated scattering y*y --+ qtj or y*y -+ gg, and nonperturbative matrix elements measuring the transitions qtj -+ AB and gg + AB. These matrix elements have been called generalized distribution amplitudes (GDAs) to emphasize their close connection to the distribution amplitudes introduced many years ago in the QCD description of exclusive hard processes [ 71. In this paper, we focus on the process yy* + pop0 which has recently been observed at LEP in the right kinematical domain [ 81. We calculate the differential cross-section of the considered processes as a function of Q2 and compare it to the experimental data.
2.
Kinematics The reaction which we study here is:
e(k) + 4)
-+ 4k’) + 40
+ pa(n)
+
p”(P2)
(1)
where the initial electron e(lc) radiates a hard virtual photon with momentum q = k - k’ ; in other words, the square of virtual photon momentum q2 = -Q2 is very large. This means that the scattered electron e(k’) is tagged. To describe reaction (l), it is useful to consider, at the same time, the sub-process of reaction (1) : e(k) + y(d)
-+ 4k’) + Pi
+ P~CPZ).
(2)
Regarding the other photon momentum qf = l-l’, we assume that, firstly, its momentum is collinear to the electron momentum I and, secondly, that equal to zero, which is a 4‘2 is approximately usual approximation when the second lepton is untagged. Let us now pass to a short discussion of kinematics in the yy* center of mass system. We adopt the z axis directed along the threedimensional vector q, and the p-meson momenta are in the (z, z)-plane. So, we write for momenta in the c.m. system :
9 = (90,0, 0, 4, m = (18, plsino, 0920-5632/s - see front matter 0 2004 Published by Elsevier B.V. doi: 10.1016/~j.nuc1p1~ysbps.2003.1 I .047
0, PICOS~)
(3)
I. R Anikin et ul./Nuclear
278
Physics B (Pmt. Suppl.) 126 (2004) 277-282
Also, we need to write down the Mandelstam S-variables for the electron-positron (1) and electron-photon (2) collisions:
Within this frame the transverse component of the transfer momentum AT = (0, AT, 0) is given by
se, = (k + E)2,
*$=--A$=
s,, = (k + q’)2.
(4
Neglecting the lepton masses, these variables can be rewritten as S ee M 2(k. I),
se, M 2(k. q’) = 22See,
(5)
where the fraction 22 defined as qh = x210 is introduced (see, for instance [ 91).
(W2--4naz)sin?8.
We now come to the parameterization of the relevant matrix elements. Keeping the terms of leading twist 2, the vector and axial correlators can be written as (see, for instance, [ lo], where the crossed case of the generalized parton distributions in a spin-l particles is considered): (Pl,Xl;P2,X2113;(0)~,~(Xn)lO)
3. Parameterization of p-matrix and their properties
(p . n) = 1.
(6)
In this basis, the pmesons can be written as
momenta
1-
(I
1+
As usual, we introduce hadronic momenta: -p;,
pl and pz
A; 2 ’ 3 2
(7)
the sum and difference of
P’” =pg
The skewedness parameter
+p’i”.
(8)
t is defined by
A en = -eP.n,
(9)
or
I+(-
lfPCOS8 2
2
In (13) and (14), the tensor structures V$(pl,p~, n) and Af$(p,,p,, n) depend on the vectors pl, p2 and n. Notice that due to the parity invariance the tensors VA2 may be written in terms of five tensor structures while the tensors A$ are linear combinations of four independent structures.
1-g. d-----
=c.
g,T,W/)V(y, case, W2)+
(10)
Notice that the relation between our definition and definition in, for instance, [ 91 reads
T
of yy* 4 pop0 subprocess
In this section, we pass to the consideration of y(q’)y*(q) + p”(pl)po(pz) subprocess. Following [ 111, the amplitude of this subprocess including the leading twist-2 terms can be written as
T,v= ,p=
(13)
Here Xi and X2 are the helicities of spin-l hadrons and 2 denotes the Fourier transformation with measure (21 = Xn,zs = 0) [ 111: $p(y) = dy e--iY Wl-ti(l-Y) PZZ (15)
4. Amplitude Ap =p;
2
elements
Let us first introduce the basis light-cone vectors. We adopt a light-cone basis consisting of two light-like vectors p and n of mass dimension 1 and -1, respectively. In other words, they obey the following conditions : p2 = n2 = 0,
(12)
@+(Y)A(Y,
~0~8, W”> 1 ,
(16)
where (17)
i. R Anikin et al. /Nuclear Physics B (Proc. Suppl.) 126 (2004) 277-282
In (16), the scalar and pseudo-scalar functions (V, A) denote the following contraction V(y,cosB,W2)
A(y,
cos 8, W”)
12
=
-Q~czz
= (1 - a2)E,2sin2 omaz.
=
zz .;(i)$T’4v.
du(e+ty -+ e+pOpO) _ a3 ,!3 1 dW2 dcos6’d# 16~ S& 0”
1- 25'e,(Q2+ W2-se,) (Q2 + W2)2
(19)
Here, in the yy* c.m. frame system, the photon polarization vectors read
]A(+,+) (~0s 0, W”) I2
4 jV(cos0, (
(20)
e+e- + e+e-pop’)
JJ
=
IA(cos
W2)j2).
(25)
summed
W2)12 = P=-(pl)Pp1pz(p2)
(26)
dy&(yl)HfPT
v(yl,cosh’,
W2)
dy&(yz)HjPP’
V(y2rcos8, W”)
where the Weizsacker-Williams defined as usual : Fww(xz)
=
I+
(I2x2
-- 1 - x2 ’
~2)~~~
= P”‘““(p1)Pp1p2(pz)
dyz~+(yz)H~'"(~z,cose,
du(e+y -+ e+pOpO) dW2 dcos8 d4 ’
3vw(x2)
8, W2)12
function
(21)
(27)
W”) yc~ Jdyl~+(yl)H~P’A(yl,co
dW2 dcos8 d$ dxz
x2
W2)j2 -I- ]A(cosO,
cross-sections
We will now concentrate on the calculation of the differential cross-section of (1). Using the equivalent photon approximation we find the expression for the corresponding cross-section :
..
=
In (25), the squared and polarization functions IV12 and IAl2 read : lV(cos8,
for the real and virtual photons, respectively.
(24)
where [A(+,+)(cosW”)~~
5. Differential
(23)
The angle Q,,, in (23) is determined by the acceptance of a lepton in the detector (see, for instance, [ 91) and the value of & is 91 GeV at LEPl and 195 GeV at LEP2. In (21), the cross-section for the subprocess can be calculated directly ; we have
The next objects of our consideration are the helicity amplitudes that are obtained from the usual amplitudes after multiplying by the photon polarization vectors &)
and the value Q’ 2 is defined as Q maz
=
279
W”),
where
FWW is The helicity amplitude after the integration cosB is implemented may be written as
Q’2(x2)
over
m2
(22)
F(+,+)(W2) =J dcose ~-$+:+~b~4W")/~. (29)
280
I. r Anikin et al. /Nucleur Physics B (Prvc. Suppl.) 126 (2004) 277-282
So, the cross-section takes the form e+e- -+ e+e-pop’) a4 1 dxzFww(xd 167~~
Finally, the cross-section (30) is expressed through these three phenomenological parameters in the following simple way:
=
-j$(e+e(
1 16x;E,4Q2 I dW2P F(+,+) (W”>-
Fww
-
2
(30)
’
Notice that the integrated amplitude F(+,+) is independent of Q2 up to logarithms. This is quite natural for the cases where the factorization theorem is applied. Besides, the exact W2 dependence of this amplitude remain unknown unless some modeling is used. However, the mean value theorem gives the possibility to reduce the three different integrals over W2 in (30) to one integration. Indeed, for our case the mean value theorem reads :
I
,jW2 pF(+>+) cw2)= Q2+W2 1
Q2 + (WI)” I
I
dW2P F(,,,)
W”>>
dW2 p F(+,+) cw2) _ (Q2 + W2)2 -
dW2P F(+,+) W”) (31) (Q2 + &2j212 I with two phenomenological parameters (WI) , (W2). Because of that the values of these parameters, generally speaking, have the same order of magnitude as Q, i.e. are not much less than Q, we need to keep (I&‘)-parameters in the prefactors of (31). Thus, we can see from (31) that it is useful to introduce a third phenomenogical parameter as W?L Cl =
I YL
dW2P F(+,+)(W2L
= &
Cl
I
dx2
1 2 16x;E,4Q2 + Q2(Q2 + W2)2)2 1
1 dW2 @F(+,+) cw2) + 2x2 E;Q2 I Q2+W2 dW2 P F(,,,) W”) (Q2 + W2)2 &2 I
(~2)
+ e+e-pop’)
(32)
where the integration over W2 runs into the limits dictated by the experiment.
2x&Q2(Q2 + (WI)“)
’
(33)
Within the analysis implemented by the L3 Collaboration, the value W is in the interval 1.1 GeV < W < 3.0GeV. Hence we are able to conclude that the phenomenological parameters (WI) and (W2) may take, generally speaking, any values inside this interval. 6. Numerical
results
and Discussion
In the previous section we obtained a simple expression for the cross-section as a function of three parameters which are (WI), (WZ) and Cr. Let us now make a fit of these phenomenological parameters in order to get the best description of experimental data. The best values of the parameters can be found by the method of least squares, x2-method, which flows from the maximum likelihood theorem (see, for instance, [ 121). As usual, the x2-sum as a function of parameters is written in the form : (34) where P = {(WI), (WZ), Cr} denotes the set of fitted parameters ; ozZp and cf” are the experimental measurements of the cross-section and its theoretical estimations ; Sai are the statistical errors. The experimental data for the cross-section of the exclusive double p” production were taken from the measurement of the L3 collaboration at LEP [ 81. Further, minimizing x2-sum in (34) with respect to the parameters P we find that the set of solutions Pmin with the confidence intervals are the following : Cl = 1.0 f 0.12 GeV2, (W2) = 1.2 f 0.08 GeV, (WI) = 2.9 f 1.8 GeV. (35)
I. R Anikin
With this the magnitude and, therefore, we have
et al. /Ntrclear
of x2 is equal to
Physics
.25
X2 - 0.25 < 1 degree of freedom The confidence intervals were defined for the case of one-standard deviation. The graphics with the experimental data and theoretical cross-section for the fitted parameters (35) are drawn on Fig. 1. The obtained value of the normalization constant Ci of order one is probably related to the energy momentum conservation, which should define the scale of GDA, like in two pions [ 131 case. Besides, notice that within L3 experiment the interval of changing for W in the yy* center of mass is fixed as 1.1 GeV < W < 3.0 GeV. Therefore, we are able to note, taking into account (35), that the confidence intervals for parameters (WI) lie into the whole available interval for W. In other words, we can infer that the theoretical crosssection depends on (WI) slightly. Indeed, despite all the three terms in (33) are participating in our analysis, the behaviour of the cross-section is mainly dictated by the third term in (33).
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7. Conclusion Data thus are in full agreement with the theoretical expectation on the Q2 behaviour of the cross section. Because of that the effective structure function F(+,+j is independent of Q2 up to the logarithms, as it was said before, our analysis of the data is a direct proof of the relevance of the partonic description of the process yy* -+ pop0 in the kinematics of the L3 experiment. Much more can be done if detailed experimental results are collected. For instance the angular dependence of the final state is a good test of the validity of the asymptotic form of the generalized distribution amplitudes. The spin structure of the final state, if elucidated, would allow to disentangle the roles of the nine generalized distribution amplitudes. The W2 behaviour of the cross section may have some interesting features. It depends much on the possible resonances which are able to couple to two p mesons. Its Fourier transform allows to have access to an impact picture of exclusive fragmentation [ 141. The p+pchannel may be calculated along the same lines. In that case a brehmstrahlung subprocess where the mesons are radiated from the lepton line must be added. The charge and spin asymmetries then come from the interference of the two processes. In that way one may also study the large contribution of 47r continuum. These items will be discussed in a forthcoming publication.
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We acknowledge useful discussions and correspondance with M. Diehl, A. Nesterenko and I. Vorobiev. REFERENCES
Figure 1. Cross-section do/dQ2[pb/GeV2] as a function of Q2 . The theoretical cross-section is plotted for the best fitted parameters which are Cl = 1 .OGeV2 , (WI) = 2.9GeV and (WZ) = 1.2 GeV. Experimental data [ 81 come from LEPl and LEP2 runs.
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