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Heavy quark impact factor at NLO
Available online at www.sciencedirect.com
Nuclear and Particle Physics Proceedings 273–275 (2016) 2743–2745 www.elsevier.com/locate/nppp
Heavy quark impact factor at NLO Michal De´ak, Grigorios Chachamis, Germ´an Rodrigo Instituto de F´ısica Corpuscular, Universitat de Val`encia – Consejo Superior de Investigaciones Cient´ıficas, Parc Cient´ıfic, E-46980 Paterna, Valencia, Spain
Abstract We present the calculation of the finite part of the heavy quark impact factor at next-to-leading logarithmic (NLx) accuracy. The final result is written in a form suitable for phenomenological studies such as the calculation of the cross-section for single heavy quark production at the LHC within the kT -factorization scheme.
1. Introduction
2. High energy factorisation
Significant developments in the last two decades in small-x physics made possible the phenomenological analysis of deep inelastic scattering (DIS) and other high energy scattering processes within the kT factorization scheme. They were mainly driven by the Balitsky-Fadin-Kuraev-Lipatov (BFKL) framework for the resummation of high center-of-mass energy logarithms at leading (Lx) [1] and next-to-leading (NLx) [2] logarithmic accuracy. An important ingredient for studying high energy scattering processes within the kT -factorisation scheme is the impact factor, a process dependent object. The impact factors for gluons and massless quarks have been calculated in Ref. [3], at NLx accuracy. This allows for the calculation of various processes with massless quarks and gluons in the initial state. The generalization to hadron-hadron collisions has also been established [4, 5, 6]. The NLx impact factor for a massive quark in the initial state has been calculated in Ref. [7]. However, the result was given in a form of a sum over an infinite number of terms. To make the result of Ref. [7] available for phenomenological studies we recalculate the NLx heavy quark impact factor in a compact and resummed form which is more suitable for numerical applications.
In the high energy limit: ΛQCD |t| s, the partonic cross-section of 2 → 2 processes factorises into the impact factors ha (k1 ) and hb (k2 ) of the two colliding partons a and b, and the gluon Green’s function Gω (k1 , k2 ) (here in Mellin space) so that the differential cross-section can be written as
http://dx.doi.org/10.1016/j.nuclphysbps.2015.10.049 2405-6014/© 2015 Elsevier B.V. All rights reserved.
dσab = d[k1 ] d[k2 ]
dω ha (k1 ) Gω (k1 , k2 ) hb (k2 ) 2πi ω ω s , × s0 (k1 , k2 )
where ω is the dual variable to the rapidity Y, and d[k] = d2+2ε k/π1+ε is the transverse space measure. The Lx quark impact factor can be expressed by a very simple formula: h(0) (k) =
π 2C F αS Nε , 2 Nc − 1 k2 μ2ε
Nε =
(4π)ε/2 , Γ(1 − ε)
and it is the same (up to a color factor) for quarks and gluons. Variables μ and ε are the renormalization scale and the dimensional regularization parameter respectively.
M. Deák et al. / Nuclear and Particle Physics Proceedings 273–275 (2016) 2743–2745
2744
3. NLx correction in an integral form The NLx impact factor can be written as a sum of the leading order impact factor and NLx correction hq (k2 ) = h(0) (k2 ) + h(1) q (k2 )
(1)
According to Ref. [7], the NLx correction for the impact factor of a heavy quark can be written as the sum of three contributions: 1 (1) (1) dz1 d[k1 ]ΔFq (z1 , k1 , k2 ) hq (k2 ) = hq,m=0 (k2 ) + 0 m + d[k1 ] αS h(0) Θm k1 , q (k1 ) K0 (k1 , k2 ) log k1 with the convention Θm k1 = θ (m − k1 ), where the θfunction is the well-known Heaviside step function and k1 = |k1 |. The first term on the right hand side of Eq.(2) is the NLx correction to the impact factor of a massless quark, which can be expressed by using the leading order impact factor h(0)(k)and the gluon Regge trajectory Γ2 (1+ε) k2 ε ω(1) (k) = − α2εS Γ(1+2ε) , μ2 1 3 +K , = h (k2 ) ω (k2 ) b0 + − ε 2 2 (2) nf π2 67 with the beta function b0 = 11 − and K = − 6 3Nc 18 6 −
h(1) (k ) q,m=0 2
(0)
(1)
5n f 9Nc .
The singular term h(1) q (k)|sing is given in [7]. The finite contribution, which is our main result, finally reads
π2 αS Nc (0) K− +1 hq (k2 )|finite = h (k2 , αS (k2 )) 1 + 2π 6 1+R + 2 log(Z) − R log(4R) − log(Z) (1 + 2R) R √ 1−Z − 3 R Li2 (Z) − Li2 (−Z) + log(Z) log 1+Z 1 1 1 log (4R) + log2 (4R) + Li2 Θk2 m + 2 2 4R + Li2 (4R) Θm k2 . (4) √ √ with R = k2 /(4m2 ) and Z = ( 1 + R + R)−1 . As in Ref. [7], we have absorbed the singularities proportional to the beta function into the running of the strong coupling α s (k) [8], for details we refer the reader to Ref. [9]. The heavy quark impact factor, as described here, will be applied to high-energy phenomenology study of the cross section for single heavy quark forward production at the LHC [10]. 5. Numerical results and conclusions
The second term on the right hand side of Eq.(2) is the NLx correction induced by the heavy quark mass m, with ΔFq (z1 , k1 , k2 ) defined in Ref. [7]. The third term comes from the introduction of the mass scale to the leading order BFKL kernel K0 (k1 , k2 ) which is defined as αS + 2ω(1) (k1 )δ[q] , − ε)μ2ε (3) δ[q] = π1+ε δ2+2ε (q) ,
αS K0 (k1 , k2 ) =
q2 Γ(1
with q = k1 + k2 . The first term on the right hand side of Eq.(3) corresponds to the real component of the BFKL kernel and the second one to the virtual corrections. 4. The analytic impact factor Collecting all the contributions, the impact factor of a heavy quark at NLx accuracy reads hq (k) = h(0) (k) + h(1) q (k) , and can be expressed in terms of a singular and a finite contribution hq (k) = h(1) q (k)|sing + hq (k)|finite .
The result in Eq.(4) gives us a possibility to proceed to a first numerical study of the size of the mass corrections to the impact factor at NLx accuracy. We have adopted the running coupling scheme as described in Refs. [8] with n f = 5 flavors. At Lx accuracy, we use a fixed value for the strong coupling αS = 0.2. In the plot in Fig. 1 we show the Lx as well as the NLx quark impact factor, the latter for two quark mass values, m = 0 and m = 5 GeV. We see that the NLx correction to the leading order impact factor for massless quark is positive and moderate for small k2 where the behavior is governed by the strong running coupling, but for most of the range of the plot the NLx correction is negative. For a non-zero quark mass, in the region k22 /m2 < 1, the overall correction is positive and large. It turns negative closely after k22 /m2 = 1, but for larger k22 /m2 , they follow the NLx massless curve as expected. To get a better quantitative picture of the behavior of the NLx corrections in the massless and massive case, it is valuable to study the ratios of the impact factors at Lx and NLx accuracy. In the first plot in Fig. 2 we can see that the relative size of the full NLx corrections for low k2 (k2 < 10 GeV) vary from more than +100%
M. Deák et al. / Nuclear and Particle Physics Proceedings 273–275 (2016) 2743–2745
finite h
0
k 2
2.5
hq k 2
down to some −20%, for similar mass values m = 4 GeV and m = 5 GeV. To estimate the magnitude of the NLx corrections induced purely by the quark mass, we plot, the second plot in Fig. 2, the ratio between the finite parts of the NLx massive and massless quark impact factor. The corrections are of the order of a 100% for the small k22 /m2 valeus and decrease as k22 /m2 is getting larger. The cusps in the curves are solely an effect of the choice of the factorization scale. As expected, in the limit k2 → ∞, the massless and massive NLx impact factors match such that their ratio approaches 1.
2745
2.0 m 4 GeV
1.5 1.0
m 5 GeV m 0
Acknowledgments
finite
GeV2
0.500
2
3
4
5 6 k 2 GeV
7
8
9
10
2.0 1.8
m 4 GeV
1.6 1.4 1.2 1.0
1.000
hq k 2
hq k 2
This work has been supported by the Research Executive Agency (REA) of the European Union under the Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet), by the Spanish Government and EU ERDF funds (grants FPA2011-23778 and CSD200700042 Consolider Project CPAN) and by GV (PROMETEUII/2013/007). GC acknowledges support from Marie Curie Actions (PIEF-GA-2011-298582). MD acknowledges support from Juan de la Cierva programme (JCI-2011-11382).
finite hq k 2 ,m0 finite
1
m 5 GeV
1
2
3
4
5 6 k 2 GeV
7
8
9
10
NLx , m 5 GeV
Figure 2: Transversal momentum distributions.
0.100 0.050
References
0.010
Lx , Α S 0.2
NLx , m 0
0.005 1
2
3
4
5 6 k 2 GeV
7
8
Figure 1: Transversal momentum distributions.
9
10
[1] L. N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338; E. A. Kuraev, L. N. Lipatov, V. S. Fadin, Phys. Lett. B 60 (1975) 50, Sov. Phys. JETP 44 (1976) 443, Sov. Phys. JETP 45 (1977) 199; Ia. Ia. Balitsky, L. N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822. [2] V. S. Fadin, L. N. Lipatov, Phys. Lett. B 429 (1998) 127 [hepph/9802290]; M. Ciafaloni, G. Camici, Phys. Lett. B 430 (1998) 349 [hep-ph/9803389]. [3] M. Ciafaloni and D. Colferai, Nucl. Phys. B 538, 187 (1999) [hep-ph/9806350]. [4] A. H. Mueller and H. Navelet, Nucl. Phys. B 282 (1987) 727. [5] A. Sabio Vera and F. Schwennsen, Nucl. Phys. B 776 (2007) 170 [hep-ph/0702158 [HEP-PH]]. [6] J. Kwiecinski, A. D. Martin, L. Motyka and J. Outhwaite, Phys. Lett. B 514 (2001) 355 [hep-ph/0105039]. [7] M. Ciafaloni and G. Rodrigo, JHEP 0005 (2000) 042 [hepph/0004033]. [8] G. Rodrigo and A. Santamaria, Phys. Lett. B 313 (1993) 441 [hep-ph/9305305]; G. Rodrigo, A. Pich and A. Santamaria, Phys. Lett. B 424 (1998) 367 [hep-ph/9707474]. [9] G. Chachamis, M. Deak and G. Rodrigo, JHEP 1312, 066 (2013) [arXiv:1310.6611]. G. Chachamis, M. Deak and G. Rodrigo, arXiv:1307.2780 [hep-ph]. [10] G. Chachamis, M. Deak, M. Hentschinski, G. Rodrigo and A. S. Vera, work in progress.
Nuclear and Particle Physics Proceedings 273–275 (2016) 2743–2745 www.elsevier.com/locate/nppp
Heavy quark impact factor at NLO Michal De´ak, Grigorios Chachamis, Germ´an Rodrigo Instituto de F´ısica Corpuscular, Universitat de Val`encia – Consejo Superior de Investigaciones Cient´ıficas, Parc Cient´ıfic, E-46980 Paterna, Valencia, Spain
Abstract We present the calculation of the finite part of the heavy quark impact factor at next-to-leading logarithmic (NLx) accuracy. The final result is written in a form suitable for phenomenological studies such as the calculation of the cross-section for single heavy quark production at the LHC within the kT -factorization scheme.
1. Introduction
2. High energy factorisation
Significant developments in the last two decades in small-x physics made possible the phenomenological analysis of deep inelastic scattering (DIS) and other high energy scattering processes within the kT factorization scheme. They were mainly driven by the Balitsky-Fadin-Kuraev-Lipatov (BFKL) framework for the resummation of high center-of-mass energy logarithms at leading (Lx) [1] and next-to-leading (NLx) [2] logarithmic accuracy. An important ingredient for studying high energy scattering processes within the kT -factorisation scheme is the impact factor, a process dependent object. The impact factors for gluons and massless quarks have been calculated in Ref. [3], at NLx accuracy. This allows for the calculation of various processes with massless quarks and gluons in the initial state. The generalization to hadron-hadron collisions has also been established [4, 5, 6]. The NLx impact factor for a massive quark in the initial state has been calculated in Ref. [7]. However, the result was given in a form of a sum over an infinite number of terms. To make the result of Ref. [7] available for phenomenological studies we recalculate the NLx heavy quark impact factor in a compact and resummed form which is more suitable for numerical applications.
In the high energy limit: ΛQCD |t| s, the partonic cross-section of 2 → 2 processes factorises into the impact factors ha (k1 ) and hb (k2 ) of the two colliding partons a and b, and the gluon Green’s function Gω (k1 , k2 ) (here in Mellin space) so that the differential cross-section can be written as
http://dx.doi.org/10.1016/j.nuclphysbps.2015.10.049 2405-6014/© 2015 Elsevier B.V. All rights reserved.
dσab = d[k1 ] d[k2 ]
dω ha (k1 ) Gω (k1 , k2 ) hb (k2 ) 2πi ω ω s , × s0 (k1 , k2 )
where ω is the dual variable to the rapidity Y, and d[k] = d2+2ε k/π1+ε is the transverse space measure. The Lx quark impact factor can be expressed by a very simple formula: h(0) (k) =
π 2C F αS Nε , 2 Nc − 1 k2 μ2ε
Nε =
(4π)ε/2 , Γ(1 − ε)
and it is the same (up to a color factor) for quarks and gluons. Variables μ and ε are the renormalization scale and the dimensional regularization parameter respectively.
M. Deák et al. / Nuclear and Particle Physics Proceedings 273–275 (2016) 2743–2745
2744
3. NLx correction in an integral form The NLx impact factor can be written as a sum of the leading order impact factor and NLx correction hq (k2 ) = h(0) (k2 ) + h(1) q (k2 )
(1)
According to Ref. [7], the NLx correction for the impact factor of a heavy quark can be written as the sum of three contributions: 1 (1) (1) dz1 d[k1 ]ΔFq (z1 , k1 , k2 ) hq (k2 ) = hq,m=0 (k2 ) + 0 m + d[k1 ] αS h(0) Θm k1 , q (k1 ) K0 (k1 , k2 ) log k1 with the convention Θm k1 = θ (m − k1 ), where the θfunction is the well-known Heaviside step function and k1 = |k1 |. The first term on the right hand side of Eq.(2) is the NLx correction to the impact factor of a massless quark, which can be expressed by using the leading order impact factor h(0)(k)and the gluon Regge trajectory Γ2 (1+ε) k2 ε ω(1) (k) = − α2εS Γ(1+2ε) , μ2 1 3 +K , = h (k2 ) ω (k2 ) b0 + − ε 2 2 (2) nf π2 67 with the beta function b0 = 11 − and K = − 6 3Nc 18 6 −
h(1) (k ) q,m=0 2
(0)
(1)
5n f 9Nc .
The singular term h(1) q (k)|sing is given in [7]. The finite contribution, which is our main result, finally reads
π2 αS Nc (0) K− +1 hq (k2 )|finite = h (k2 , αS (k2 )) 1 + 2π 6 1+R + 2 log(Z) − R log(4R) − log(Z) (1 + 2R) R √ 1−Z − 3 R Li2 (Z) − Li2 (−Z) + log(Z) log 1+Z 1 1 1 log (4R) + log2 (4R) + Li2 Θk2 m + 2 2 4R + Li2 (4R) Θm k2 . (4) √ √ with R = k2 /(4m2 ) and Z = ( 1 + R + R)−1 . As in Ref. [7], we have absorbed the singularities proportional to the beta function into the running of the strong coupling α s (k) [8], for details we refer the reader to Ref. [9]. The heavy quark impact factor, as described here, will be applied to high-energy phenomenology study of the cross section for single heavy quark forward production at the LHC [10]. 5. Numerical results and conclusions
The second term on the right hand side of Eq.(2) is the NLx correction induced by the heavy quark mass m, with ΔFq (z1 , k1 , k2 ) defined in Ref. [7]. The third term comes from the introduction of the mass scale to the leading order BFKL kernel K0 (k1 , k2 ) which is defined as αS + 2ω(1) (k1 )δ[q] , − ε)μ2ε (3) δ[q] = π1+ε δ2+2ε (q) ,
αS K0 (k1 , k2 ) =
q2 Γ(1
with q = k1 + k2 . The first term on the right hand side of Eq.(3) corresponds to the real component of the BFKL kernel and the second one to the virtual corrections. 4. The analytic impact factor Collecting all the contributions, the impact factor of a heavy quark at NLx accuracy reads hq (k) = h(0) (k) + h(1) q (k) , and can be expressed in terms of a singular and a finite contribution hq (k) = h(1) q (k)|sing + hq (k)|finite .
The result in Eq.(4) gives us a possibility to proceed to a first numerical study of the size of the mass corrections to the impact factor at NLx accuracy. We have adopted the running coupling scheme as described in Refs. [8] with n f = 5 flavors. At Lx accuracy, we use a fixed value for the strong coupling αS = 0.2. In the plot in Fig. 1 we show the Lx as well as the NLx quark impact factor, the latter for two quark mass values, m = 0 and m = 5 GeV. We see that the NLx correction to the leading order impact factor for massless quark is positive and moderate for small k2 where the behavior is governed by the strong running coupling, but for most of the range of the plot the NLx correction is negative. For a non-zero quark mass, in the region k22 /m2 < 1, the overall correction is positive and large. It turns negative closely after k22 /m2 = 1, but for larger k22 /m2 , they follow the NLx massless curve as expected. To get a better quantitative picture of the behavior of the NLx corrections in the massless and massive case, it is valuable to study the ratios of the impact factors at Lx and NLx accuracy. In the first plot in Fig. 2 we can see that the relative size of the full NLx corrections for low k2 (k2 < 10 GeV) vary from more than +100%
M. Deák et al. / Nuclear and Particle Physics Proceedings 273–275 (2016) 2743–2745
finite h
0
k 2
2.5
hq k 2
down to some −20%, for similar mass values m = 4 GeV and m = 5 GeV. To estimate the magnitude of the NLx corrections induced purely by the quark mass, we plot, the second plot in Fig. 2, the ratio between the finite parts of the NLx massive and massless quark impact factor. The corrections are of the order of a 100% for the small k22 /m2 valeus and decrease as k22 /m2 is getting larger. The cusps in the curves are solely an effect of the choice of the factorization scale. As expected, in the limit k2 → ∞, the massless and massive NLx impact factors match such that their ratio approaches 1.
2745
2.0 m 4 GeV
1.5 1.0
m 5 GeV m 0
Acknowledgments
finite
GeV2
0.500
2
3
4
5 6 k 2 GeV
7
8
9
10
2.0 1.8
m 4 GeV
1.6 1.4 1.2 1.0
1.000
hq k 2
hq k 2
This work has been supported by the Research Executive Agency (REA) of the European Union under the Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet), by the Spanish Government and EU ERDF funds (grants FPA2011-23778 and CSD200700042 Consolider Project CPAN) and by GV (PROMETEUII/2013/007). GC acknowledges support from Marie Curie Actions (PIEF-GA-2011-298582). MD acknowledges support from Juan de la Cierva programme (JCI-2011-11382).
finite hq k 2 ,m0 finite
1
m 5 GeV
1
2
3
4
5 6 k 2 GeV
7
8
9
10
NLx , m 5 GeV
Figure 2: Transversal momentum distributions.
0.100 0.050
References
0.010
Lx , Α S 0.2
NLx , m 0
0.005 1
2
3
4
5 6 k 2 GeV
7
8
Figure 1: Transversal momentum distributions.
9
10
[1] L. N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338; E. A. Kuraev, L. N. Lipatov, V. S. Fadin, Phys. Lett. B 60 (1975) 50, Sov. Phys. JETP 44 (1976) 443, Sov. Phys. JETP 45 (1977) 199; Ia. Ia. Balitsky, L. N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822. [2] V. S. Fadin, L. N. Lipatov, Phys. Lett. B 429 (1998) 127 [hepph/9802290]; M. Ciafaloni, G. Camici, Phys. Lett. B 430 (1998) 349 [hep-ph/9803389]. [3] M. Ciafaloni and D. Colferai, Nucl. Phys. B 538, 187 (1999) [hep-ph/9806350]. [4] A. H. Mueller and H. Navelet, Nucl. Phys. B 282 (1987) 727. [5] A. Sabio Vera and F. Schwennsen, Nucl. Phys. B 776 (2007) 170 [hep-ph/0702158 [HEP-PH]]. [6] J. Kwiecinski, A. D. Martin, L. Motyka and J. Outhwaite, Phys. Lett. B 514 (2001) 355 [hep-ph/0105039]. [7] M. Ciafaloni and G. Rodrigo, JHEP 0005 (2000) 042 [hepph/0004033]. [8] G. Rodrigo and A. Santamaria, Phys. Lett. B 313 (1993) 441 [hep-ph/9305305]; G. Rodrigo, A. Pich and A. Santamaria, Phys. Lett. B 424 (1998) 367 [hep-ph/9707474]. [9] G. Chachamis, M. Deak and G. Rodrigo, JHEP 1312, 066 (2013) [arXiv:1310.6611]. G. Chachamis, M. Deak and G. Rodrigo, arXiv:1307.2780 [hep-ph]. [10] G. Chachamis, M. Deak, M. Hentschinski, G. Rodrigo and A. S. Vera, work in progress.