Multiple interactions and generalized parton distributions
aa r X i v : . [ h e p - ph ] J u l Multiple interactions and generalized partondistributions
Markus Diehl ∗ Deutsches Elektronen-Synchroton DESY, 22603 Hamburg, GermanyE-mail: [email protected]
Multiple parton interactions in a single proton-proton collision are expected to play an importantrole for many observables at LHC. To a large part their phenomenological description relies onrather simple and physically intuitive assumptions. We report on an investigation of multiple hardinteractions in QCD, which aims at identifying to which extent this simple description arises fromtheory and where it needs to be extended. An approximate connection with generalized partondistributions may help elucidate some aspects of multiple-interaction dynamics.
XVIII International Workshop on Deep-Inelastic Scattering and Related Subjects, DIS 2010April 19-23, 2010Firenze, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ultiple interactions and generalized parton distributions
Markus Diehl
1. Introduction
In proton-proton collisions at very high energies one can have events in which several par-tons in one proton scatter on partons in the other proton and produce particles of large transversemomentum or large mass. The effects of such multiparton interactions average out in sufficientlyinclusive observables, which can be described by conventional factorization formulae that involvea single hard scattering. Multiple interactions do however change the structure of the final state andmay be important for many analyses at LHC [1]. A brief review on the subject can be found in [2].Phenomenological estimates of multiparton dynamics rely on models that are physically in-tuitive but rather simplified, whereas a systematic description in QCD remains a challenge. Thepresent contribution reports on some steps towards this goal. The discussion is limited to tree-levelgraphs, which already exhibit important features. Loop corrections, in particular soft gluon ex-change, add a further layer of complexity and are essential for determining if and in which formfactorization holds. For details and further discussion we refer to [3].
2. Cross section formula and multiparton distributions
An example process where multiparton interactions contribute is the production of two elec-troweak gauge bosons ( W , Z or g ∗ with large virtuality) in kinematics where the transverse mo-menta of the bosons are small compared with their mass or virtuality. Since we are interested in thedetails of the final state, we keep the cross section differential in the transverse boson momenta. Forthe corresponding process with a single boson, a powerful theory in terms of transverse-momentumdependent parton densities has been developed [4], which one may hope to generalize to the caseof multiple hard scattering.Figure 1a shows a graph with two hard-scattering subprocesses. The transverse momentumof each produced boson is the sum of the transverse momenta of a quark and an antiquark. Withtwo partons emitted by each proton the transverse parton momenta are in general not equal in thescattering amplitude and its conjugate, as illustrated in the figure. Momentum conservation implies r + ¯ r = for the differences between the parton momenta in the amplitude and its conjugate.Performing a Fourier transform w.r.t. r and ¯ r one finds that their Fourier conjugate variables areequal, y = ¯ y . q q (a) k − r ¯ k − ¯ r ¯ k + ¯ rk + rk + r k − r ¯ k − ¯ r ¯ k + ¯ rp ¯ p q q (b) Figure 1:
Graphs for the production of two gauge bosons by double (a) or single (b) hard scattering. ultiple interactions and generalized parton distributions Markus Diehl
Making the usual kinematic approximations in the hard-scattering subprocesses (i.e. neglect-ing small momentum components compared with large ones) one obtains d s(cid:213) i = dx i d ¯ x i d q i (cid:12)(cid:12)(cid:12)(cid:12) Fig. 1 a = S (cid:229) a , a = q , D q , d q ¯ a , ¯ a = ¯ q , D ¯ q , d ¯ q ˆ s a ¯ a ( ˆ s = x ¯ x s ) ˆ s a ¯ a ( ˆ s = x ¯ x s ) × (cid:20) (cid:213) i = Z d k i d ¯ k i d ( ) ( q i − k i − ¯ k i ) (cid:21) Z d y F a , a ( x i , k i , y ) F ¯ a , ¯ a ( ¯ x i , ¯ k i , y ) (2.1)with S = S = s a i ¯ a i denotes the hard-scattering cross section for single-boson production, and the kinematic variables are s = ( p + ¯ p ) , x i = ( q i ¯ p ) / ( p ¯ p ) and ¯ x i = ( q i p ) / ( ¯ p p ) . The definition of double-parton distributions closely resem-bles the one for the transverse-momentum dependent distribution of a single quark [4]. For thetwo-quark distribution in the proton with momentum p we have F a , a ( x i , k i , y ) = (cid:20) (cid:213) i = Z dz − i d z i ( p ) e ix i z − i p + − i z i k i (cid:21) p + Z dy − × (cid:10) p (cid:12)(cid:12) ¯ q (cid:0) − z (cid:1) G a q (cid:0) z (cid:1) ¯ q (cid:0) y − z (cid:1) G a q (cid:0) y + z (cid:1)(cid:12)(cid:12) p (cid:11)(cid:12)(cid:12)(cid:12) z + = z + = y + = , (2.2)where we use light-cone coordinates u ± = ( u ± u ) / √ u . One may regard k i as the “average” transverse momentum of each quark and y as their “average” transverse distancefrom each other, where the “average” refers to the scattering amplitude and its conjugate. Re-garding the transverse coordinates F ( x i , k i , y ) has the structure of a Wigner distribution [5], whichdepends on both momentum and position variables for each particle. (This does not contradict theuncertainty principle since Wigner distributions are not probability densities.) A similar applicationof this concept has been discussed for generalized parton distributions in [6].Integrating (2.1) over the transverse boson momenta q and q one obtains d s(cid:213) i = dx i d ¯ x i (cid:12)(cid:12)(cid:12)(cid:12) Fig. 1 a = S (cid:229) a , a = q , D q , d q ¯ a , ¯ a = ¯ q , D ¯ q , d ¯ q ˆ s a ¯ a ( ˆ s = x ¯ x s ) ˆ s a ¯ a ( ˆ s = x ¯ x s ) × Z d y F a , a ( x i , y ) F ¯ a , ¯ a ( ¯ x i , y ) . (2.3)The transverse-momentum integrated distribution F a , a ( x i , y ) = R d k d k F a , a ( x i , k i , y ) maybe interpreted as the probability density for finding two quarks with momentum fractions x and x at a relative transverse distance y in the proton. The form (2.3) has long been known (see e.g. [7, 8])and underlies most phenomenological models for multiparton interactions.For each quark there are three relevant Dirac matrices in (2.2), G q = g + , G D q = g + g , G j d q = i s j + g with j = , , (2.4)which respectively project on unpolarized, longitudinally polarized and transversely polarizedquarks. Note that polarized two-parton distributions appear even in an unpolarized proton, sincethey describe spin correlations between the two partons. For small but comparable x and x one3 ultiple interactions and generalized parton distributions Markus Diehl may well have sizeable spin correlations between the two quarks (which are close in rapidity for x ∼ x ), even if there is little correlation between the polarizations of a quark and the proton (whichare far apart in rapidity). The relevance of such correlations in multiparton interactions was pointedout already in [8] but has to our knowledge never been included in phenomenological estimates.Note that if parton spin correlations are sizeable they can have a strong impact on observ-ables. For the production of two gauge bosons one can easily see that the product F D q , D q F D ¯ q , D ¯ q of longitudinal spin correlations enters the cross section with the same weight as the unpolarizedterm F q , q F ¯ q , ¯ q . One also finds that product F d q , d q F d ¯ q , d ¯ q of transverse spin correlations give rise toa cos ( j ) modulation in the angle j between the decay planes of the two bosons and thus affectsthe distribution of final-state particles.The formulae given so far have ignored the color structure of the multiparton distributions. Thequark lines with momenta k ± r in Fig. 1a can couple to a color singlet (as in single-parton distri-butions) but they can also couple to a color octet, provided that the lines with momenta k ± r arecoupled to a color-octet as well. Such color-octet distributions contribute to the multiple-scatteringcross section, as was already pointed out in [8]. A more detailed discussion will be given in [3].
3. Power behavior
It is easy to determine the power behavior of the cross section formula (2.1) for two hardscatters. The hard-scattering cross sections ˆ s behave like 1 / Q , where Q ∼ q ∼ q denotes thesize of the large squared mass or virtuality of the gauge bosons. With transverse momenta q ∼ q of generic hadronic size L we find d s(cid:213) i = dx i d ¯ x i d q i (cid:12)(cid:12)(cid:12)(cid:12) Fig. 1 a ∼ Q L , (3.1)where we have used that the two-parton distributions scale like F ∼ / L and that the typicaltransverse distance y between the partons is of order 1 / L . The same power behavior as in (3.1)is obtained for the case where both bosons are produced in a single hard scattering, as shown inFig. 1b. For the cross section differential in the transverse boson momenta, multiple hard interac-tions are therefore not power suppressed.The situation changes when one integrates over q and q . In the double-scattering mechanismboth transverse momenta are restricted to be of size L , but for a single hard scattering one has | q + q | ∼ L whereas the individual transverse momenta can be as large as Q . Because of thisphase space effect one has d s(cid:213) i = dx i d ¯ x i (cid:12)(cid:12)(cid:12)(cid:12) Fig. 1 a ∼ L Q , d s(cid:213) i = dx i d ¯ x i (cid:12)(cid:12)(cid:12)(cid:12) Fig. 1 b ∼ Q . (3.2)In the transverse-momentum integrated cross section, multiple hard scattering is therefore onlya power correction. This is required for the validity of the usual factorization formulae, whichdescribe only the single-scattering contribution. 4 ultiple interactions and generalized parton distributions Markus Diehl
4. Connection with generalized parton distributions
A simple ansatz for modeling two-parton distributions is to write them as a product of single-parton distributions, thus neglecting correlations between the two partons. This provides a startingpoint for phenomenology, even though one may not expect such an approximation to be very pre-cise. A way to implement this ansatz for the distributions F ( x i , k i , y ) is to insert a sum (cid:229) X | X i h X | over all physical states between the two bilinear operators in (2.2). If one assumes that the protonstate is dominant in this sum, one has (cid:10) p (cid:12)(cid:12) (cid:0) ¯ q x q x (cid:1) (cid:0) ¯ q x q x (cid:1)(cid:12)(cid:12) p (cid:11) ≈ Z d p ′ + p ′ + d p ′ ( p ) (cid:10) p (cid:12)(cid:12) ¯ q x q x (cid:12)(cid:12) p ′ (cid:11) (cid:10) p ′ (cid:12)(cid:12) ¯ q x q x (cid:12)(cid:12) p (cid:11) , (4.1)where the subscripts x , x indicate the momentum fraction associated with each field and spinorindices have been omitted for brevity. After the Fourier transform in (2.2), momentum conser-vation imposes p ′ + = p + but one still has p ′ = p . The two-parton distribution F ( x i , k i , y ) isthus approximated by a product of two generalized parton distributions with zero skewness x .For transverse-momentum integrated distributions one obtains a simple representation F ( x i , y ) = R d b f ( x , b ) f ( x , b + y ) in terms of impact parameter dependent parton densities [9].The same method can be applied to the color-octet distributions mentioned above. Rearrangingthe quark operators as (cid:0) ¯ q x l a q x (cid:1) (cid:0) ¯ q x l a q x (cid:1) = − (cid:0) ¯ q x q x (cid:1) (cid:0) ¯ q x q x (cid:1) − (cid:0) ¯ q x q x (cid:1) (cid:0) ¯ q x q x (cid:1) , (4.2)where l a are the Gell-Mann matrices, one can insert proton states between the operators in paren-theses on the r.h.s. For the first term this leads to generalized parton distributions with nonzero skewness x , since the fields coupled to a color singlet carry different longitudinal momenta.Although it is only approximate, the connection with generalized parton distributions that canbe measured in exclusive reactions — pointed out already in [10] — will hopefully help to betterunderstand at least some features of multiple hard interactions. References [1] H. Jung et al. , arXiv:0903.3861 [hep-ph];S. Alekhin et al. , hep-ph/0601012 and hep-ph/0601013.[2] T. Sjöstrand, P. Z. Skands, JHEP (2004) 053 [hep-ph/0402078].[3] M. Diehl, D. Ostermeier, A. Schäfer, in preparation.[4] J. C. Collins, D. E. Soper, Nucl. Phys.
B193 (1981) 381;X. D. Ji, J. P. Ma, F. Yuan, Phys. Lett.
B597 (2004) 299 [hep-ph/0405085];J. C. Collins, T. C. Rogers, A. M. Stasto, Phys. Rev.
D77 (2008) 085009 [arXiv:0708.2833 [hep-ph]].[5] M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, Phys. Rept. (1984) 121.[6] A. V. Belitsky, X. D. Ji, F. Yuan, Phys. Rev.
D69 (2004) 074014 [hep-ph/0307383].[7] N. Paver, D. Treleani, Z. Phys.
C28 (1985) 187.[8] M. Mekhfi, Phys. Rev.
D32 (1985) 2371; Phys. Rev. D (1985) 2380.[9] M. Burkardt, Int. J. Mod. Phys. A18 (2003) 173 [hep-ph/0207047].[10] L. Frankfurt, M. Strikman, C. Weiss, Phys. Rev.
D69 (2004) 114010 [hep-ph/0311231].(2004) 114010 [hep-ph/0311231].