Proton structure in high-energy high-multiplicity p-p collisions
FFew-Body Systems manuscript No.
FBSY-D-16-00038R1 DOI: 10.1007/s00601-016-1091-3
Stanis(cid:32)law D. G(cid:32)lazek · Patryk Kubiczek
Proton structure in high-energy high-multiplicityp-p collisions
Received: 25 January 2016 / Accepted: 15 March 2016
Abstract
A few-body proton image, expected to be derivable from QCD in the renormalizationgroup procedure for effective particles, is used within the Monte Carlo Glauber model to calculate theanisotropy coefficients in the initial collision-state of matter in high-energy high-multiplicity proton-proton interaction events. We estimate the ridge-like correlations in the final hadronic state by assumingtheir proportionality to the initial collision-state anisotropy. In our estimates, some distinct few-bodyproton structures appear capable of accounting for the magnitude of p-p ridge effect, with potentiallydiscernible differences in dependence on multiplicity.
Keywords high-energy proton-proton collisions · two-particle correlations · collective flow · protonstructure · renormalization group Protons resist precise theoretical description of their internal dynamics in the Minkowski space-timefor a long time by now. The simplest such picture, which is provided by the constituent quark modelused to classify hadrons, is not precisely derived from QCD. The theory itself uses the Euclidean-spacetechniques that do not easily yield any real space-time image. In these circumstances, it is of interestto note that high-energy high-multiplicity proton-proton ( pp ) collisions may shed new light on theissue of proton structure. Namely, the numerous products in such collisions exhibit collective behaviorthat appears dependent on the initial state of colliding proton matter and the latter depends on theproton structure. Thus, the correlations among products in high-energy high-multiplicity pp collisionsmay report on the proton structure.In particular, the CMS [1] and ATLAS [2] collaborations reported the collective flow in pp collisionsthat resembles the one observed in heavy-ion collisions [3; 4; 5]. Several authors discussed such flows in pp collisions [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] and we follow the insights of Ref. [6] in order toestimate the extent to which the high-energy high-multiplicity pp events are sensitive to the model ofproton structure. Following the approach of Ref. [6] and the parameter choice such as in Ref. [15], usedhere, means making a strong assumption that the parton medium produced in the overlap region of pp collision at the LHC has similar hydrodynamical properties as that in heavy ion collisions at RHIC. Theindividual proton structures we consider are motivated by the general features of the renormalizationgroup procedure for effective particles (RGPEP) in quantum field theory [19; 20]. S. D. G(cid:32)lazekInstitute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, PolandE-mail: [email protected]. KubiczekFaculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, (cid:32)Lojasiewicza 11, 30-348Krak´ow, PolandE-mail: [email protected] a r X i v : . [ h e p - ph ] M a y We find that, the effective picture of a quark and diquark with a gluon flux between them producesa different dependence of eccentricity and triangularity on multiplicity than the three-quark picturewith a star-like junction made of gluons does. According to this finding, the recent data for high-energy high-multiplicity events suggest a significant star-like gluon junction component in the protonstructure. Our analysis also indicates a need for assessing the adequacy of the linear relationship usedby us between asymmetries in the initial collision state, such as the eccentricity or triangularity, andthe final state correlations in high-multiplicity events, such as the elliptic flow. pp collisions As mentioned in Sec. 1, the ridge-effect in pp scattering can be described using the hydrodynamicevolution of the asymmetric state of matter that results from one proton’s quark and gluon distribu-tion suddenly colliding with another’s. The asymmetric state is meant to evolve according to laws ofhydrodynamics until it eventually turns into the detected particles that emerge through hadronizationin the final state, in which they exhibit the angular correlation over a long-range in rapidity, calledthe ridge. The final state ridge-like correlations, such as the elliptic flow, are thus related to the initialstage of pp collision whose nature depends on the proton structure. One of the key issues is thus howto describe the proton structure using QCD in the Minkowski space-time. Fig. 1
Proton structure described using effective quarks of size s that is very much smaller than, smaller andcomparable with the constituent quark size s c , with the single large circle indicating the volume available foreffective gluons irrespective of their corresponding size, see Ref. [19]. Conceptually, we approach this issue using the RGPEP [19; 20], which is a candidate for providingthe mathematical tools for describing protons as bound states of effective quarks and gluons of specificsize s . The size parameter s plays the role of an arbitrary renormalization-group scale that can beadjusted to the physical process one wants to accurately describe in simplest possible terms. Thiscondition means choosing the right variables for grasping the essence of physics most economicallyfrom the computational point of view. The scale dependence of the proton structure expected in theRGPEP is illustrated in Fig. 1 and the corresponding examples of the color structure are shown inFig. 2. The expectation is based on the scale-dependent features of effective Hamiltonians, which implythe possibility that a relativistic bound-state eigenvalue problem can be equivalently written in termsof a few-body problem for sizable effective quarks and gluons instead of an infinite combination of barepoint-like quarks and gluons of canonical QCD. Hence, the RGPEP provides the scheme in which thedistribution of matter in proton can be imagined in terms of wave functions, or probability distributionsfor the effective quarks and gluons of size s .The few-body picture of protons in QCD suggested by the RGPEP allows us to preliminarily modelthe proton structure in terms of shapes illustrated by two typical examples in Fig. 3 [21], knowing thatsuch models can in future be verified in theory. We ask if the ridge effect can phenomenologicallydistinguish between the effective configurations.We consider three types of configurations. The proton quark-diquark configuration, denoted by I and shown on the left-hand side of Fig. 3, is motivated by Refs. [17; 18]. It is a superposition of a few Fig. 2
Color structure of effective quarks for two values of the RGPEP scale parameter s ∼ s c and s < s c inFig. 1. Pions are meant to couple to nucleons in the constituent quark picture only at the nucleon boundaries,where color is not neutralized, and for smaller values of s the quarks form more localized objects with the grayarea indicating the volume available for effective gluons of a similar size to the quarks (drawing from Ref. [22]). Gaussians that represent a quark, a diquark and gluons forming a tube in between. The three-quarkconfiguration, denoted by Y and shown on the right-hand side of Fig. 3, is motivated by Ref. [19]. It isa superposition of Gaussians that represent three quarks and additional gluons forming the Y-shapedjunction. The shape of Y configuration is kept fixed. In addition, we consider the Gaussian fluctuatingthree-quark configuration, denoted by G-f , which is the same as the Y configuration but with the shapeparameters generated according to Gaussian probability distributions. Details of all configurations weconsider are available in Refs. [21; 22]. Fig. 3
Effective constituent configurations of typical size s in proton: on the left is the quark-diquark con-figuration labeled in the text as I and on the right is the three-quark configuration with a star-junction builtfrom gluons labeled in the text by Y . pp collision and the final state correlations Following Ref. [6], we adapt a simple Glauber model, widely used for modelling high energy nuclearcollisions [23], to describe the density of binary partonic collisions in scattering of two systems A and B , n coll ( x, y ; b, Σ A , Σ B ) = σ gg (cid:90) ∞−∞ dz ρ (cid:18) x − b , y, z ; Σ A (cid:19) (cid:90) ∞−∞ dz (cid:48) ρ (cid:18) x + b , y, z (cid:48) ; Σ B (cid:19) , (1) which is a function of the coordinates x and y in the plane transverse to the colliding beams, the impactparameter b and the varying parameters Σ that identify the proton structure and its orientation inspace. The coefficient σ gg denotes a parton-parton scattering cross-section, in our estimates on theorder of 4 mb, and ρ denotes the three-dimensional parton distribution described in Sec. 2.Eccentricity (cid:15) and triangularity (cid:15) in the initial stage of pp collision are calculated using theformula [24] (cid:15) n = (cid:113) { s n cos( nφ ) } + { s n sin( nφ ) } { s n } , (2)in which the curly brackets denote the expectation value { f ( x, y ) } = (cid:82) dx dy f ( x, y ) n coll ( x, y ; b, Σ A , Σ B ) (cid:82) dx dy n coll ( x, y ; b, Σ A , Σ B ) , (3)and coordinates are parameterized as x = s cos φ , y = s sin φ . The number of collisions in an event is N coll ( b, Σ A , Σ B ) = (cid:90) dx dy n coll ( x, y ; b, Σ A , Σ B ) (4)and the cross-section density in the impact parameter plane is σ ( b, Σ A , Σ B ) = 1 − (cid:20) − N coll ( b, Σ A , Σ B ) N g (cid:21) N g . (5)The total pp cross-section is thus σ pp = (cid:90) ∞ πb db (cid:90) P ( Σ A ) d Σ A (cid:90) P ( Σ B ) d Σ B σ ( b, Σ A , Σ B ) , (6)where P ( Σ ) is the probability density for proton configuration Σ . For any quantity Q , its expectationvalue in many collisions is (cid:104) Q (cid:105) = 1 σ pp (cid:90) ∞ πb db (cid:90) P ( Σ A ) d Σ A (cid:90) P ( Σ B ) d Σ B σ ( b, Σ A , Σ B ) Q ( b, Σ A , Σ B ) . (7)We used randomly oriented proton configurations in the Monte Carlo generation of about 3 · eventsfor each proton model and estimated the averaged eccentricity (cid:15) and triangularity (cid:15) in the resultingsamples. In our estimates, σ gg ∼ . σ pp ∼
60 mb [25] required the number of scatteringpartons N g = 9 ± N = αN coll andreproducing the value (cid:104) N (cid:105) = 30 [26] for charged particles by choosing α = 8 ±
3. Our minimum biasresults for eccentricity and triangularity are shown in Fig. 4It is visible in Fig. 4 that the initial stage of pp collision is characterized by different multiplicitydependence of the asymmetries for different proton structures. In collisions of quark-diquark ( II )and Gaussian-fluctuating ( G-f ) structures, the asymmetries decrease with multiplicity above about N = 100, while in the collisions of tripod three-quark configurations ( YY ) the initial stage asymmetriespersist or even increase above N = 100.In order to relate the eccentricity and triangularity to data, we note that the observable multiplicitydistributions in transverse momentum p T and pseudorapidity η are conventionally written as d Nd p T dη = (cid:40) ∞ (cid:88) n =1 v n ( p T , η ) cos [ n ( φ − Φ RP )] (cid:41) d N πp T dp T dη , (8)where the reaction plane angle Φ RP for colliding spherically symmetric distributions is illustrated inFig. 5. In the actual events the angles Φ RP are determined from the particle distributions.Assuming that the averaged minimal bias elliptic flow parameter v is proportional to the minimalbias eccentricity parameter (cid:15) with coefficient order 0.3 [15], we obtain v ∼ .
11, 0.14 and 0.09 for theproton models II , YY and G-f , respectively, while data indicates v in the range 0.04-0.08 [5]. Multiplicity N . . . . . . . . R M S ecce n tr i c i t y p h (cid:15) i G-fIIYY N . . . . . . . . R M S tr i a n g u l a r i t y p h (cid:15) i G-fIIYY
Fig. 4
Average eccentricity and triangularity obtained using Eq. (7) for three different types of protonstructure in pp collisions. Green curves labeled II correspond to collisions of protons in configuration I inFig. 3, and the red curves labeled YY correspond to collisions in configurations Y in Fig. 3. The blue lineslabeled G-f correspond to the Gaussian fluctuating three-quark configuration in which the Y -type protonconfigurations appear with different shape parameters according to a Gaussian probability distribution [21].Note the distinct multiplicity dependence in the case of YY configurations. Fig. 5
View of the initial stage of pp collision along the beam for illustration of Eq. (8). The average valueof the elliptic flow coefficient v ( p T , η ), denoted by v , is assumed to be proportional to the eccentricity (cid:112) (cid:104) (cid:15) (cid:105) obtained in Eq. (7) with a coefficient on the order of 0.3 [15]. The magnitudes of v resulting from differentproton models are displayed with quark-diquark = II , triangular = YY . More generally, taking into account that the initial-stage asymmetry parameters (cid:15) n in Eq. (2) andthe final-state coefficients v n in Eq. (8) are small, and assuming that a set of averaged coefficients v n with different n depends approximately linearly on the set of averaged parameters (cid:15) n , one can infer thedependence of v n on multiplicity N from the dependence of (cid:15) n on N . Accordingly, Fig. 4 suggests thatthe averaged elliptic flow v and higher correlations in high-energy high-multiplicity pp events mayindicate which configurations of effective quarks and gluons in the proton structure are most likely tooccur. Namely, only the YY configurations lead to (cid:15) and (cid:15) that do not fall off for N exceeding about100. Actually, recent data [2] from ATLAS Collaboration for pp collisions with √ s ∼
13 TeV show that v is stable for large N . According to our analysis, this finding favors the configuration YY .It should be observed that the long-range, near-side angular correlations in pp collisions at LHCenergies can also be studied in terms of multiparton interactions [27]. Interactions of four partons tofour partons and beyond allow for linking the ridge effect to the models [28] and theory [29] of doubleparton distributions and their light-front analysis [30]. A unified approach is hence greatly desired. Simple model estimates suggest that the correlations among final-state particles in high-energy high-multiplicity pp collisions are sensitive to the proton structure. It is not excluded that future sophis- ticated calculations will identify some spatial features of protons through precise interpretation ofexperimental data on these correlations. The multidimensional linear relationship between initial-stateasymmetries and final-state flow parameters that we assume in our estimates requires verification, e.g.,in the hydrodynamic model. If confirmed, it would provide an efficient way for studying the protonstructure through correlations in high-multiplicity pp collisions using the universal matrices determinedby the nature of assumed underlying collision dynamics. Interpreting the most recent LHC data onmultiplicity dependence of the elliptic flow coefficient v using our simple estimates, suggests thatprotons may often occur in the configuration of three effective quarks connected by a Y -shaped gluonjunction. References
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