BFKL resummation effects in gamma* gamma* to rho rho Page: 4 of 23
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The general solution for arbitrary values of t is more involved  and will not be discussed
here. We only want to stress that it has a simple expression at the Born level,
A = is 2 k2 (kr - k)b(kr- k), (6)
k (r - k)
which has recently been evaluated in Ref. . The special case r2 = 0 of Eq. (6) can be
readily obtained from Eq. (3). This relation will be studied in Section 2.2. The impact
factor l*L - PL is given at t = tmin by [3
i (k, r = 0) = CabQi1J dz zz O(z) (2 2
Jo + k2+mn
where mi = Qizf (with z = 1 - z), O(z) is the distribution amplitude of the meson which
is here given by its asymptotic form O(z) = 6zz, and'
cab = 1()672a 2 f, (8)
(2w)2 s/22N '
where f, is the p0-meson decay constant.
Inserting (7) and (8) in (3) we get
fW(r2 = 0) = 47a aem NN - 1 01 f2 dzi ziz #(zi) jdz2 z2f2 #(z2) (9)
x O-O(Vn) f k m2 k2 +m2 ) 2w] [J k' m2 - k'2 +m2) kf 2w]
Let us denote the expressions in the square brackets as I(z, v) and I(z, -v) respectively.
It is straightforward to put the propagators on a common denominator and show that
I(z, v) = - (m12)-92 F (2- iv) F (-2 + iv)
= - (Q1) 2 (zz)-+av F (2 - iv) F (-2 + iv) . (10)
Next, the z integrals can also be done. We have
dz zz #z) I(z, v) (2- F - iv) F (-2 + iv) dz 6 (zf)i-i"
= -6 q2 -2-2i (Q2)-2+iF (3 -iv) F (-1 iv) 2 ~ v
F (2 + iv)
since the integral over z is just the definition of the Euler beta function. Finally this
yields the Mellin transform
f (r2 = 0) = 9 72a ,aem N c 1 1 23/2+yv ( d -3/2-( )u
N, p j dvw - wv(v)
F2 (2 - iv) F2 (2 + iv) F (-2 - iv) F (-2 + iv)
F (2-i) F (2+ )
'The first factor 1/(27r)2 differs from  and comes from the convention for impact factors.
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Enberg, R.; Pire, B.; Szymanowski, L. & Wallon, S. BFKL resummation effects in gamma* gamma* to rho rho, article, August 11, 2005; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc892501/m1/4/?rotate=90: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.