- Email: [email protected]
Baryon Stopping in pA Collisions
Nuclear Physics A 783 (2007) 573c–576c
Baryon Stopping in pA Collisions Javier L. Albacete ∗ and Yuri V. Kovchegov Department of Physics, Ohio State University, Columbus, OH 43210, USA Abstract We calculate the inclusive small-x valence quark production cross section in protonnucleus collisions at high energies. The calculation is performed in the framework of the Color Glass Condensate formalism. First we calculate the cross section in the quasi-classical approximation and then we include the the effects of double logarithmic reggeon evolution and leading logarithmic gluon evolution in the obtained cross section.
1
Introduction
In this work we consider the problem of baryon stopping, the net transfer of baryon number along a large rapidity gap in nuclear collisions, from a purely perturbative perspective. In our approach the carriers of baryon number are the energetic valence quarks entering the collision, that loose much of their longitudinal momenta in the collision and are driven towards the central rapidity region via hard gluon emissions. 2
Quasi-classical calculation.
We are interested in calculating the soft valence quark spectrum in protonnucleus collisions at high energies. In what follows we will assume that the proton is moving ultrarelativistically in the light-cone ’plus’ direction, whereas ∗ Corresponding author. Email addresses: [email protected] (Javier L. Albacete), [email protected] (Yuri V. Kovchegov). 1 This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-05ER41337.
0375-9474/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.nuclphysa.2006.11.119
574c
J.L. Albacete, Y.V. Kovchegov / Nuclear Physics A 783 (2007) 573c–576c
a nucleon in the nucleus is fast moving in the light-cone ’minus’ direction. The calculation is done in the framework of light-cone perturbation theory and in the light-cone of the proton, A+ = 0. As usual at high energies we stick to the eikonal approximation, such that fast partons propagate through the nucleus keeping their transverse coordinates fixed. In the quasi-classical approximation the number of gluons exchanged between each nucleon in the nucleus and the rest of the system is limited to two, such that the parameter for the resummation of the multiple rescatterings is αs2 A1/3 . Corrections to this approximation are considered in the next section. We divide the cross section for soft quark production into two terms: the first contribution, (a), corresponds to produced valence quark coming from the proton, while the second one, (b), corresponds to valence quark coming from one of the nucleus in the nucleus via interaction with a soft gluon in the proton wavefunction (see Fig. 1): dσ dσ a dσ b = + . (1) d2 k dy d2 k dy d2 k dy Our result for these two contributions is:
¯ s Nc y−Y /2 |k| 2 2 2 ik(x−y) x · y 1 α dσ a √ d xd yd be = e d2 k dy (2π)2 2π s x2 y 2 2 Q2 sq
× e−(x−y)
ln(1/|x−y|Λ)/4
2 Q2 sq
− e−x
ln(1/|x|Λ)/4
− e−y
2 Q2 sq
ln(1/|y|Λ)/4
+1 ,
(2)
and αs CF e−y+Y /2 2 2 2 ik(x−y) x · y 1 1 2¯ dσ b √ d xd yd be = CF 2 2 2 2 d k dy (2π) π 1 − Nc s|k| x y (x − y)2
2 Q2 sq
× e−(x−y)
ln(1/|x−y|Λ)/4
2 Q2 sg
− e−(x−y)
ln(1/|x−y|Λ)/4
,
(3)
where k and y are the transverse momentum and rapidity of the produced valence quark. Eqs (2) and (3) admit a rather clear interpretation: they consist in the convolution of the squared soft quark (gluon for Eq. (3)) emission amplitude from the proton with the propagation through the nucleus of the resultant system after the emission, given by the eikonal Glauber-Mueller factors in the r.h.s of both equations. Eq. (3) also accounts for the absorption of the emitted gluon by one of the fast valence quarks in the nucleus, which is kicked out from its light-cone trajectory towards the central rapidity region.
3
Quantum Evolution.
At smaller values of Bjorken-x quantum corrections to the quasi-classical cross section become important due to the emission of extra gluons. In the large-Nc limit, the inclusion of quantum effects in the first contribution to the cross
575c
J.L. Albacete, Y.V. Kovchegov / Nuclear Physics A 783 (2007) 573c–576c
proton z
+ x
y
x
...
...
...
...
y
...
...
nucleus
A
B
Fig. 1. Two of the diagrams contributing to dσ a) (left) and dσ b) (right).
section is achieved replacing Eq. (2) by dσ a ¯ s Nc α 2 2 2 1 α , α ) = d x d y d z1 d2 z eik(x−y) r(z 0 , z 1 , α1 ; z, α) (z 1 10 d2 k dy (2π)2 2π α1 (x − z) · (y − z) × N (x, z, y) + N (y, z, y) − N (x, y, y) , (4) |x − z|2 |y − z|2 where r(z 0 , z 1 , α1 ; z, α)/α1 has the meaning of the probability density of finding a soft valence quark at transverse coordinate z carrying longitudinal momentum fraction greater than or equal to α in the wave function of a dipole z 10 with the initial longitudinal momentum fraction carried by the quark being α1 . The quantity r accounts for the emission of extra gluons with rapidity larger than the one of the produced valence quark and it obeys the following linear evolution equation in the large-Nc limit [1]: 2 /z 2 } 21 dα d2 z2 α ¯ s α1 min{1,z01 r(z 1 , z 2 , α ; z, α). 2 2π α α z12 (5) On the other hand, the replacement of the Glauber-Mueller propagators in Eq. (2) by the corresponding rapidity dependent dipole scattering amplitudes in Eq. (4), N (xi , xj , Y ), accounts for the radiation of gluons softer than the produced valence quark. The energy evolution of N is given by the non-linear Balitsky-Kovchegov equation [2]:
r(z 0 , z 1 , α1 ; z, α) = δ 2 (z−z 1 ) +
∂N01 x201 αs N c 2 d x [N02 + N12 − N01 − N02 N12 ] = 2 2 ∂Y 2 π2 x20 x221
(6)
An analogous discussion for the inclusion of quantum effects in dσ b) can be found in [3]. 4
Cronin enhancement.
A more quantitative study of the nuclear effects in the small-x valence quark production mechanism is achieved by introducing the nuclear modification fac-
576c
J.L. Albacete, Y.V. Kovchegov / Nuclear Physics A 783 (2007) 573c–576c pA
pp
RpA
dσ tor, RpA (y, kT ) = ddσ 2 k dy /(A d2 k dy ). The results for RpA are shown in Fig. 2 for central rapidity and different values of rapidity towards the proton fragmentation region. They have been obtained via numerical solution of Eqs. (4)-(6). At central rapidity RpA shows a Cronin-type peak at kT Qs . This enhancement is quickly wiped out by quantum evolution for more forward rapidities towards the proton fragmentation region, analogously to what was predicted and experimentally confirmed for small-x gluon production[4–6].
2 1.8
ΔY=0,1,2,3,4
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 1
2
3
4
5
6
7
8
9
10
k T (GeV)
Fig. 2. Evolution of RpA (kT , y) for valence quarks with rapidity. The initial condition corresponds to Q2sA = 1 GeV2 for a nucleus and Q2sp = Λ2 = 0.1 GeV2 for a proton.
Therefore, the experimental determination of the nuclear modification factor of the net baryon yield produced in pA collisions could serve as an additional check for the small-x/CGC dynamics as well as to disentangle the perturbative and non-perturbative contributions to the phenomenon of baryon stopping.
References [1] K. Itakura, Y. V. Kovchegov, L. McLerran and D. Teaney, Nucl. Phys. A 730, 160 (2004) [arXiv:hep-ph/0305332]. [2] I.Balitsky, Nucl. Phys. B 463, 99 (1996)[arXiv:hep-ph/9509348]; Y.V. Kovchegov, Phys. Rev. D60, 034008 (1999)[arXiv:hep-ph/9509348]. [3] Javier L. Albacete and Yuri V. Kovchegov, arXiv:hep-ph/0605053. [4] J. L. Albacete, N. Armesto, A. Kovner, C. A. Salgado and U. A. Wiedemann, Phys. Rev. Lett. 92, 082001 (2004) [arXiv:hep-ph/0307179]. [5] D. Kharzeev, Y. V. Kovchegov and K. Tuchin, Phys. Rev. D 68, 094013 (2003) [arXiv:hep-ph/0307037]. [6] I. Arsene et al. [BRAHMS Collaboration], Phys. Rev. Lett. 93, 242303 (2004) [arXiv:nucl-ex/0403005].
Baryon Stopping in pA Collisions Javier L. Albacete ∗ and Yuri V. Kovchegov Department of Physics, Ohio State University, Columbus, OH 43210, USA Abstract We calculate the inclusive small-x valence quark production cross section in protonnucleus collisions at high energies. The calculation is performed in the framework of the Color Glass Condensate formalism. First we calculate the cross section in the quasi-classical approximation and then we include the the effects of double logarithmic reggeon evolution and leading logarithmic gluon evolution in the obtained cross section.
1
Introduction
In this work we consider the problem of baryon stopping, the net transfer of baryon number along a large rapidity gap in nuclear collisions, from a purely perturbative perspective. In our approach the carriers of baryon number are the energetic valence quarks entering the collision, that loose much of their longitudinal momenta in the collision and are driven towards the central rapidity region via hard gluon emissions. 2
Quasi-classical calculation.
We are interested in calculating the soft valence quark spectrum in protonnucleus collisions at high energies. In what follows we will assume that the proton is moving ultrarelativistically in the light-cone ’plus’ direction, whereas ∗ Corresponding author. Email addresses: [email protected] (Javier L. Albacete), [email protected] (Yuri V. Kovchegov). 1 This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-05ER41337.
0375-9474/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.nuclphysa.2006.11.119
574c
J.L. Albacete, Y.V. Kovchegov / Nuclear Physics A 783 (2007) 573c–576c
a nucleon in the nucleus is fast moving in the light-cone ’minus’ direction. The calculation is done in the framework of light-cone perturbation theory and in the light-cone of the proton, A+ = 0. As usual at high energies we stick to the eikonal approximation, such that fast partons propagate through the nucleus keeping their transverse coordinates fixed. In the quasi-classical approximation the number of gluons exchanged between each nucleon in the nucleus and the rest of the system is limited to two, such that the parameter for the resummation of the multiple rescatterings is αs2 A1/3 . Corrections to this approximation are considered in the next section. We divide the cross section for soft quark production into two terms: the first contribution, (a), corresponds to produced valence quark coming from the proton, while the second one, (b), corresponds to valence quark coming from one of the nucleus in the nucleus via interaction with a soft gluon in the proton wavefunction (see Fig. 1): dσ dσ a dσ b = + . (1) d2 k dy d2 k dy d2 k dy Our result for these two contributions is:
¯ s Nc y−Y /2 |k| 2 2 2 ik(x−y) x · y 1 α dσ a √ d xd yd be = e d2 k dy (2π)2 2π s x2 y 2 2 Q2 sq
× e−(x−y)
ln(1/|x−y|Λ)/4
2 Q2 sq
− e−x
ln(1/|x|Λ)/4
− e−y
2 Q2 sq
ln(1/|y|Λ)/4
+1 ,
(2)
and αs CF e−y+Y /2 2 2 2 ik(x−y) x · y 1 1 2¯ dσ b √ d xd yd be = CF 2 2 2 2 d k dy (2π) π 1 − Nc s|k| x y (x − y)2
2 Q2 sq
× e−(x−y)
ln(1/|x−y|Λ)/4
2 Q2 sg
− e−(x−y)
ln(1/|x−y|Λ)/4
,
(3)
where k and y are the transverse momentum and rapidity of the produced valence quark. Eqs (2) and (3) admit a rather clear interpretation: they consist in the convolution of the squared soft quark (gluon for Eq. (3)) emission amplitude from the proton with the propagation through the nucleus of the resultant system after the emission, given by the eikonal Glauber-Mueller factors in the r.h.s of both equations. Eq. (3) also accounts for the absorption of the emitted gluon by one of the fast valence quarks in the nucleus, which is kicked out from its light-cone trajectory towards the central rapidity region.
3
Quantum Evolution.
At smaller values of Bjorken-x quantum corrections to the quasi-classical cross section become important due to the emission of extra gluons. In the large-Nc limit, the inclusion of quantum effects in the first contribution to the cross
575c
J.L. Albacete, Y.V. Kovchegov / Nuclear Physics A 783 (2007) 573c–576c
proton z
+ x
y
x
...
...
...
...
y
...
...
nucleus
A
B
Fig. 1. Two of the diagrams contributing to dσ a) (left) and dσ b) (right).
section is achieved replacing Eq. (2) by dσ a ¯ s Nc α 2 2 2 1 α , α ) = d x d y d z1 d2 z eik(x−y) r(z 0 , z 1 , α1 ; z, α) (z 1 10 d2 k dy (2π)2 2π α1 (x − z) · (y − z) × N (x, z, y) + N (y, z, y) − N (x, y, y) , (4) |x − z|2 |y − z|2 where r(z 0 , z 1 , α1 ; z, α)/α1 has the meaning of the probability density of finding a soft valence quark at transverse coordinate z carrying longitudinal momentum fraction greater than or equal to α in the wave function of a dipole z 10 with the initial longitudinal momentum fraction carried by the quark being α1 . The quantity r accounts for the emission of extra gluons with rapidity larger than the one of the produced valence quark and it obeys the following linear evolution equation in the large-Nc limit [1]: 2 /z 2 } 21 dα d2 z2 α ¯ s α1 min{1,z01 r(z 1 , z 2 , α ; z, α). 2 2π α α z12 (5) On the other hand, the replacement of the Glauber-Mueller propagators in Eq. (2) by the corresponding rapidity dependent dipole scattering amplitudes in Eq. (4), N (xi , xj , Y ), accounts for the radiation of gluons softer than the produced valence quark. The energy evolution of N is given by the non-linear Balitsky-Kovchegov equation [2]:
r(z 0 , z 1 , α1 ; z, α) = δ 2 (z−z 1 ) +
∂N01 x201 αs N c 2 d x [N02 + N12 − N01 − N02 N12 ] = 2 2 ∂Y 2 π2 x20 x221
(6)
An analogous discussion for the inclusion of quantum effects in dσ b) can be found in [3]. 4
Cronin enhancement.
A more quantitative study of the nuclear effects in the small-x valence quark production mechanism is achieved by introducing the nuclear modification fac-
576c
J.L. Albacete, Y.V. Kovchegov / Nuclear Physics A 783 (2007) 573c–576c pA
pp
RpA
dσ tor, RpA (y, kT ) = ddσ 2 k dy /(A d2 k dy ). The results for RpA are shown in Fig. 2 for central rapidity and different values of rapidity towards the proton fragmentation region. They have been obtained via numerical solution of Eqs. (4)-(6). At central rapidity RpA shows a Cronin-type peak at kT Qs . This enhancement is quickly wiped out by quantum evolution for more forward rapidities towards the proton fragmentation region, analogously to what was predicted and experimentally confirmed for small-x gluon production[4–6].
2 1.8
ΔY=0,1,2,3,4
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 1
2
3
4
5
6
7
8
9
10
k T (GeV)
Fig. 2. Evolution of RpA (kT , y) for valence quarks with rapidity. The initial condition corresponds to Q2sA = 1 GeV2 for a nucleus and Q2sp = Λ2 = 0.1 GeV2 for a proton.
Therefore, the experimental determination of the nuclear modification factor of the net baryon yield produced in pA collisions could serve as an additional check for the small-x/CGC dynamics as well as to disentangle the perturbative and non-perturbative contributions to the phenomenon of baryon stopping.
References [1] K. Itakura, Y. V. Kovchegov, L. McLerran and D. Teaney, Nucl. Phys. A 730, 160 (2004) [arXiv:hep-ph/0305332]. [2] I.Balitsky, Nucl. Phys. B 463, 99 (1996)[arXiv:hep-ph/9509348]; Y.V. Kovchegov, Phys. Rev. D60, 034008 (1999)[arXiv:hep-ph/9509348]. [3] Javier L. Albacete and Yuri V. Kovchegov, arXiv:hep-ph/0605053. [4] J. L. Albacete, N. Armesto, A. Kovner, C. A. Salgado and U. A. Wiedemann, Phys. Rev. Lett. 92, 082001 (2004) [arXiv:hep-ph/0307179]. [5] D. Kharzeev, Y. V. Kovchegov and K. Tuchin, Phys. Rev. D 68, 094013 (2003) [arXiv:hep-ph/0307037]. [6] I. Arsene et al. [BRAHMS Collaboration], Phys. Rev. Lett. 93, 242303 (2004) [arXiv:nucl-ex/0403005].