Multiperipheral final states in crowded twin-jet events at the LHC
MMultiperipheral final statesin crowded twin-jet events at the LHC
N. Bethencourt de Le´on , G. Chachamis , A. Sabio Vera , Instituto de F´ısica Te´orica UAM/CSIC, Nicol´as Cabrera 15, E-28049 Madrid, Spain. Laborat´orio de Instrumenta¸c˜ao e F´ısica Experimental de Part´ıculas (LIP),Av. Prof. Gama Pinto, 2, P-1649-003 Lisboa, Portugal. Theoretical Physics Department, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain.
February 11, 2021
Abstract
The 13 TeV run of the LHC has provided a unique opportunity to explore multi-jet final stateswith unprecedented accuracy. An interesting region for study is that of events where one jet istagged in the forward direction and another one in the backward direction and a plethora of lowenergy mini-jets populate the possibly large rapidity span in between them. Since the numberof these events is very high, it is possible to introduce stringent constraints on the transversemomentum of the two leading jets which can be kept in small windows not very different fromeach other, defining what we call “twin jets”. The associated “crowd” of mini-jets can also havea restricted span in transverse momentum. The study of these events for a fixed multiplicity isan ideal playground to investigate different models of multi-particle production in hadron-hadroncollisions. We set up an exploratory analysis by using an ancient model of Chew and Pignotti todescribe the gross features one can expect for the structure of single and double differential-in-rapidity cross sections and for particle-particle rapidity correlations when the longitudinal phasespace completely decouples from the transverse degrees of freedom.
A fascinating open problem in Quantum Chromodynamics (QCD) is the correct understanding ofmulti-particle events generated in the interaction at high center-of-mass energies of leptons withhadrons or hadrons with hadrons since its theoretical description is quite cumbersome when higher or-der quantum corrections are taken into account. Experimental data on multi-particle hadroproductionin the last 5-6 decades have continuously reinforced the concept that the final state particles tend tomerge into correlated clusters [1]. The investigation of exclusive quantities and correlations betweenfinal state particles has been offering important information about the strong force in high-energyinteractions studied in collider experiments, well before the advent of QCD. Evidently, high-ordercorrelations among emitted particles have been found in all types of hadron collisions as manifesta-tions of multiplicity fluctuations. Furthermore, particle correlations summarize important propertiesof jets without being too sensitive to missing soft particles in the jet [2] (see also the discussion in theintroduction of Ref. [3]).There are many different effective approaches to multi-particle hadroproduction which are mainlybased on the resummation of the leading contributions to the scattering amplitudes associated tothese type of events (e.g. DGLAP [4–8], BFKL [9–13], CCFM [14, 15], Linked Dipole Chain [16–18],Lund Model [19]). The correct understanding of these issues is very important since it is related to thereduction of the theoretical uncertainty in the parton distribution functions of hadrons at small values1 a r X i v : . [ h e p - ph ] F e b f Bjorken x and also to Standard Model backgrounds for new physics searches with high multiplicityat the Large Hadron Collider (LHC).The latest 13 TeV run of the LHC has generated a large amount of data which is needed tounderstand in detail. In the present work we would like to highlight a type of final state configurationswhich are very important to fix the range of applicability of multi-particle production models. Theycorrespond to events with several jets where the two outermost in rapidity jets are found one in theforward and one in the backward regions and can be cleanly tagged since their transverse momentum isrelatively large. In order to pin down events with low momentum transfer we propose to focus on eventswhere these two jets have similar transverse momentum in what we call “twin-jet” configurations.Together with these twin-jets a set of low energy “mini-jets”, with their p T also constrained withina limited window and which we can refer to as the “crowd”, are also present. This corresponds toa subset of the so-called Mueller-Navelet jets [20]. We require a very similar p T for the twin-jets(although not within overlapping range) to enforce a de facto symmetry on their influence upon thecrowd mini-jets.One of the advantages of the 13 TeV run of the LHC is that these constrained events are numerousand their statistical analysis can reach a good level of accuracy. To improve their study at future LHCruns with higher energies it will be important to have some periods of dedicated low luminosity runsavailable.Our task in the present work is to investigate some of the gross features which can be expected inthese processes. Since the most striking characteristics of the final states in multiperipheral modelsand within the cluster concept are present in the rapidity space, with their origin stemming from thedecoupling of transverse coordinates from longitudinal ones due to kinematical reasons, we will focusthis first exploratory analysis on differential cross sections and particle-particle correlations in rapidityspace.We introduce the method of analysis in Section 2, together with the description of single differ-ential distributions. An expansion on a finite basis of Chebyshev polynomials is introduced whosecorresponding coefficients could be compared with a similar analysis of the LHC data. In Section 3double differential distributions and their corresponding finite expansions on products of Chebyshevpolynomials are discussed in detail. These are compared with equivalent ones in the case of completelyindependent particle production to fix the region in phase space where particle-particle correlationscan be expected. These correlations are explored in more detail in Section 4. In Section 5 we furthermotivate the study of the Chew-Pignotti model by comparing it with a collinear limit of the BFKLequation and discuss the kinematical cuts needed at the LHC to measure the proposed observables.After this we draw some Conclusions. In a simplified analysis we work with a Chew-Pignotti type of multiperipheral model [21] followingthe discussion by DeTar [22] which will allow us to generate some analytic results that can be easilycompared to the experimental data . In these types of models the dependence on transverse coordi-nates factorizes from the rapidity dependence and we can write the cross section for the productionof N + 2 particles (the “2” refers to the twin-jets which in each event are considered to have rapidities ± Y ) in the simple form σ N +2 = α N +2 (cid:90) Y N +1 (cid:89) i =1 dz i δ (cid:32) Y − N +1 (cid:88) s =1 z s (cid:33) = α N +2 (cid:90) Y − Y dy N (cid:90) y N − Y dy N − · · · (cid:90) y − Y dy (cid:90) y − Y dy = α ( αY ) N N ! . (1) For a review on multiperipheral models and the cluster concept in multiple hadron production see Ref. [1] andreferences therein. σ total = (cid:80) ∞ N =0 σ N +2 = α e αY whose growth with Y can betamed only by introducing a non trivial dynamics in transverse space, something beyond the simplescope of the work here presented. We assign a rapidity y l , with l = 0 , . . . , N + 1, to each of thefinal-state particles. At y = − Y and y N +1 = Y we position our “twin-jets” with a simplified jetvertex equal to α , the strong coupling constant . The set of mini-jets with l = 1 , . . . N is what wedenote as “the crowd” for which y l = − Y + (cid:80) lj =1 z j .More than in the total cross section, we are interested in a qualitative description of the differentialdistributions for each final state in events with fixed multiplicity. Multiplicity in our discussion isdefined as the number of mini-jets (or even jets) in the crowd assuming that an IR safe jet clusteringalgorithm has been applied to the final state particles. One can argue that the multiplicity is nota unique number for a final state and that it depends on the lower transverse momentum cutoffand the resolution radius in the rapidity-azimuthal angle plane one considers for the jet algorithm.Nevertheless, we expect that our discussion in the following holds as long as there is a well definedmechanism that decides the multiplicity of a final state. This is actually a strong statement in itselffrom the simple model we are considering. It implies that if an event which was initially classifiedas having multiplicity N + 2 complies with the N + 2 differential distributions then it will continueto comply with the N + 2 differential distributions assuming that in the latter case a different setof parameters is chosen for the jet clustering algorithm which results in a shifted number of crowdmini-jets from N to N . Evidently, the crowd mini-jets contribute to the N + 2 particle productioncross section with a differential distribution in rapidity which can be derived from Eq. (1) and is ofthe form dσ ( l ) N +2 dy l = α N +2 (cid:90) Y N +1 (cid:89) i =1 dz i δ (cid:32) Y − N +1 (cid:88) s =1 z s (cid:33) δ y l + Y − l (cid:88) j =1 z j = α N +2 (cid:90) Y y l dy N (cid:90) y N y l dy N − · · · (cid:90) y l +2 y l dy l +1 (cid:90) y l − Y dy l − · · · (cid:90) y − Y dy (cid:90) y − Y dy = α N +2 (cid:0) Y − y l (cid:1) N − l ( N − l )! (cid:0) y l + Y (cid:1) l − ( l − . (2)In the very large multiplicity limit this converges to an asymptotic Poisson distribution as can beseen, e.g. , in the region y (cid:39) − Y with y = (cid:0) λN − (cid:1) Y wherelim N →∞ (cid:18) N − l − (cid:19) (cid:18) − λN (cid:19) N − l (cid:18) λN (cid:19) l − = e − λ λ l − ( l − . (3)Considering the limit l → y l → − Y in Eq. (2) we can obtain a normalized universal distributionfor each N when plotted versus 2 y/Y . We show it for multiplicity seven in Fig. 1 (left). The notationjet i =1 , ,...,N is introduced for jets ordered with rapidities y < y < · · · < y N . It can be seen that theyreflect the characteristic cluster structure in the multiperipheral region of phase space. Each of thesenormalized y -distributions spans an area of N and has a maximum at y = l − N − N − Y where its valueis (cid:18) N − l − (cid:19) ( l − l − ( N − N − ( N − l ) N − l . (4)These maxima are shown for increasing multiplicities in Fig. 1 (right). We would like to make clear at this point that we choose to work with limits y = − Y and y N +1 = Y to treatthe forward and backward rapidity direction in a symmetric way. One could in principle also work with limits y = 0and y N +1 = Y . It does not mean that an event in experimental data with e.g. y = 0 . y N +1 = 6 . y = − . y N +1 = 3 . - y jet Y total N o r m a li z e d va l u es y - distributions per jet in events with multiplicity 7jet jet jet jet jet jet jet � � �� � � �� � � � �� � � � � �� � � � � � �� � � � � � � �� � � � � � � � �� � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � � � - - y jet Y total N o r m a li z e d va l u es Maxima of y - distributions per jet in events with multiplicity N Figure 1:
Rapidity distributions for each of the jets in a final state with seven mini-jets (left). Their maximaare indicated by the symbol ○ (right). The positions of the y -distribution maxima in configurations withmultiplicity N are marked by N ○ (right). It is possible to make more concrete predictions to be eventually compared with the experimentaldata if we use the following expansion of Eq. (2) in terms of a finite number of Chebyshev polynomials T n (this is a standard procedure, see, e.g. [23]): dσ ( l ) N +2 dy l = α N +2 (cid:18) Y (cid:19) N − (1 − x l ) N − l ( N − l )! (1 + x l ) l − ( l − α N +2 (cid:18) Y (cid:19) N − N − (cid:88) s =0 C ( l ) N +2 ,s T s ( x l ) , (5)where x l = 2 y l /Y . The coefficients C ( l ) N +2 ,s = 2 − δ s π (cid:90) − dx l (cid:112) − x l T s ( x l ) (1 − x l ) N − l ( N − l )! (1 + x l ) l − ( l − y : C (1)5 , = C (1)5 , = − C (1)5 , = y : C (2)5 , = C (2)5 , = 0 C (2)5 , = − y : C (3)5 , = C (3)5 , = 1 C (3)5 , = For multiplicity 4 + 2: y : C (1)6 , = C (1)6 , = − C (1)6 , = C (1)6 , = − y : C (2)6 , = C (2)6 , = − C (2)6 , = − C (2)6 , = y : C (3)6 , = C (3)6 , = C (3)6 , = − C (3)6 , = − y : C (4)6 , = C (4)6 , = C (4)6 , = C (4)6 , = As a final example, with 5 + 2, we find 4 : C (1)7 , = C (1)7 , = − C (1)7 , = C (1)7 , = − C (1)7 , = y : C (2)7 , = C (2)7 , = − C (2)7 , = − C (2)7 , = C (2)7 , = − y : C (3)7 , = C (3)7 , = 0 C (3)7 , = − C (3)7 , = 0 C (3)7 , = y : C (4)7 , = C (4)7 , = C (4)7 , = − C (4)7 , = − C (4)7 , = − y : C (5)7 , = C (5)7 , = C (5)7 , = C (5)7 , = C (5)7 , = These are predictions associated to the clustering structure of the final state stemming from this simpleanalysis. It will be interesting to compare them with similar ones obtained from different models ofmulti-particle production.
Let us now derive similar expressions for double differential rapidity distributions for pairs of orderedin rapidity jets, i.e. d σ ( l,m ) N +2 dy l dy m = α N +2 (cid:90) Y N +1 (cid:89) i =1 dz i δ (cid:32) Y − N +1 (cid:88) s =1 z s (cid:33) δ y l + Y − l (cid:88) j =1 z j δ (cid:32) y m + Y − m (cid:88) k =1 z k (cid:33) = α N +2 (cid:0) Y − y l (cid:1) N − l ( N − l )! ( y l − y m ) l − m − ( l − m − (cid:0) y m + Y (cid:1) m − ( m − . (7)The expansion in the Chebyshev basis takes now place via pairs of polynomials: d σ ( l,m ) N +2 dy l dy m = α N +2 (cid:18) Y (cid:19) N − (1 − x l ) N − l ( N − l )! ( x l − x m ) l − m − ( l − m − x m ) m − ( m − α N +2 (cid:18) Y (cid:19) N − N − (cid:88) r,s =0 D ( l,m ) N +2 ,r,s T r ( x l ) T s ( x m ) (8)with coefficients D ( l,m ) N +2 ,r,s = (2 − δ r )(2 − δ s ) π ( N − l )!( l − m − m − × (cid:90) − dx l dx m (1 − x l ) N − l T r ( x l ) (1 + x m ) m − T s ( x m ) (cid:112) − x l (cid:112) − x m ( x l − x m ) l − m − . (9)The two-dimensional gradient of these distributions is zero at the point ( x l , x m ) max n = (cid:16) l − − nn − , m − nn − (cid:17) which is where they have a maximum value.In the case with multiplicity 2+2 we get as the only non-zero component D (2 , , , = 1 and we canformally write α d σ (2 , dy dy = 1 = T ( x ) T ( x ). For multiplicity 3+2 there is more structure:Fig. 2; ( x , x ) max n =3 = ( − , − D (2 , , , = 1 D (2 , , , = − x , x ) max n =3 = (1 , D (3 , , , = 1 D (3 , , , = 1Fig. 2; ( x , x ) max n =3 = (1 , − D (3 , , , = 1 D (3 , , , = − - - - y Y y Y Multiplicity 3 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 3 + - Jet 1 vs Jet 3
Figure 2:
Left: Double differential cross section α Y d σ (2 , dy dy = 1 − x = T ( x ) T ( x ) − T ( x ) T ( x ) (therelated distribution by x → − x is α Y d σ (3 , dy dy = 1 + x = T ( x ) T ( x ) + T ( x ) T ( x ) ). Right: Doubledifferential cross section α Y d σ (3 , dy dy = x − x = T ( x ) T ( x ) − T ( x ) T ( x ) . In both x L = 2 y L /Y . Fig. 3; ( x , x ) max n =4 = ( − , − D (2 , , , = D (2 , , , = − D (2 , , , = Fig. 3; ( x , x ) max n =4 = (1 , D (4 , , , = D (4 , , , = 1 D (4 , , , = Fig. 3; ( x , x ) max n =4 = (0 , − D (3 , , , = − D (3 , , , = 1 D (3 , , , = − D (3 , , , = − D (3 , , , = 1Fig. 3; ( x , x ) max n =4 = (1 , D (4 , , , = − D (4 , , , = 1 D (4 , , , = − D (4 , , , = − D (4 , , , = 1Fig. 4; ( x , x ) max n =4 = (1 , − D (4 , , , = D (4 , , , = D (4 , , , = − D (4 , , , = Fig. 4; ( x , x ) max n =4 = (0 , D (3 , , , = 1 D (3 , , , = − D (3 , , , = 1 D (3 , , , = − - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 3
Figure 3:
Left: Double differential cross section α Y d σ (2 , dy dy = (1 − x ) = T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) (the related distribution generated by x → − x is α Y d σ (4 , dy dy = (1 + x ) = T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) ). Right: Double differential cross section α Y d σ (3 , dy dy = (1 − x ) ( x − x ) = − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) (the related distribution generated by x → − x , x → − x is α Y d σ (4 , dy dy =( x − x ) (1 + x ) = − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) ). In both x L = 2 y L /Y . - - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 4 - - - - y Y y Y Multiplicity 4 + - Jet 2 vs Jet 3
Figure 4:
Left: Double differential cross section α Y d σ (4 , dy dy = ( x − x ) = T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) . Right: Double differential cross section α Y d σ (3 , dy dy =(1 + x ) (1 − x ) = T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) . In both x L = 2 y L /Y . x , x ) max n =5 = ( − , − D (2 , , , = D (2 , , , = − D (2 , , , = D (2 , , , = − Fig. 5; ( x , x ) max n =5 = (1 , D (5 , , , = D (5 , , , = D (5 , , , = D (5 , , , = Fig. 5; ( x , x ) max n =5 = (cid:0) − , − (cid:1) : D (3 , , , = D (3 , , , = − D (3 , , , = D (3 , , , = D (3 , , , = − D (3 , , , = Fig. 5; ( x , x ) max n =5 = (cid:0) , (cid:1) : D (4 , , , = D (4 , , , = − D (4 , , , = 1 D (4 , , , = − D (4 , , , = D (4 , , , = − Fig. 6; ( x , x ) max n =5 = (cid:0) − , − (cid:1) : D (3 , , , = − D (3 , , , = D (3 , , , = − D (3 , , , = D (3 , , , = − D (3 , , , = 1 D (3 , , , = − Fig. 6; ( x , x ) max n =5 = (cid:0) , (cid:1) : D (5 , , , = − D (5 , , , = D (5 , , , = − D (5 , , , = − D (5 , , , = − D (5 , , , = 1 D (5 , , , = Fig. 6; ( x , x ) max n =5 = (cid:0) , − (cid:1) : D (4 , , , = D (4 , , , = − D (4 , , , = D (4 , , , = − D (4 , , , = D (4 , , , = − D (4 , , , = D (4 , , , = D (4 , , , = − Fig. 6; ( x , x ) max n =5 = (cid:0) , − (cid:1) : D (5 , , , = D (5 , , , = − D (5 , , , = D (5 , , , = D (5 , , , = − D (5 , , , = D (5 , , , = D (5 , , , = − D (5 , , , = Fig. 7; ( x , x ) max n =5 = (1 , − D (5 , , , = D (5 , , , = D (5 , , , = − D (5 , , , = − D (5 , , , = D (5 , , , = − Fig. 7; ( x , x ) max n =5 = (cid:0) , − (cid:1) : D (4 , , , = − D (4 , , , = D (4 , , , = − D (4 , , , = − D (4 , , , = 2 D (4 , , , = − D (4 , , , = − D (4 , , , = with the corresponding figures being Figs. 5, 6 and 7. In order to measure the degree of correlation among different mini-jets in the final state it is customaryto evaluate the quantity R N +2 ( x l , x m ) = σ N +2 d σ ( l,m ) N +2 dy l dy m dσ ( l ) N +2 dy l dσ ( m ) N +2 dy m − N N ! ( N − m )!( l − l − m − x l − x m ) l − m − (1 + x l ) l − (1 − x m ) N − m − , (10)where Y > y l > y m > l > m and x J = 2 y J /Y . This compares the double differential cross sectionsto the totally uncorrelated case where these are obtained by simply multiplying two single differentialcross sections corresponding to each the jets in the chosen pair. Before discussing Eq. (10) we haveplotted the latter products in the form α − N +2) (cid:18) Y (cid:19) − N ) dσ ( l ) N +2 dy l dσ ( m ) N +2 dy m = (1 − x l ) N − l ( N − l )! (1 + x l ) l − ( l − − x m ) N − m ( N − m )! (1 + x m ) m − ( m − x l , x m ) max n = (cid:16) l − − nn − , m − n − n − (cid:17) . Therefore, the lower order examples readFig. 8: ( x , x ) max n =3 = (0 , − x , x ) max n =3 = (1 , x , x ) max n =3 = (1 , − x , x ) max n =4 = (cid:0) − , − (cid:1) ; ( x , x ) max n =4 = (cid:0) , (cid:1) ;Fig. 9: ( x , x ) max n =4 = (cid:0) , − (cid:1) ; ( x , x ) max n =4 = (cid:0) , − (cid:1) ;Fig. 10: ( x , x ) max n =4 = (1 , − x , x ) max n =4 = (cid:0) , − (cid:1) ;8 - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 5 + - Jet 2 vs Jet 3
Figure 5:
Left: Double differential cross section α Y d σ (2 , dy dy = (1 − x ) = T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) (the related distribution with x → − x is α Y d σ (5 , dy dy = (1 + x ) = T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) ). Right: Double differen-tial cross section α Y d σ (3 , dy dy = (1 + x ) (1 − x ) = T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) (the related distibution with x → − x and x → − x is α Y d σ (4 , dy dy = (1 + x ) (1 − x ) = T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) ). In both x L = 2 y L /Y .. Fig. 11: ( x , x ) max n =5 = (cid:0) − , − (cid:1) ; ( x , x ) max n =5 = (cid:0) , (cid:1) ;Fig. 11: ( x , x ) max n =5 = (cid:0) , − (cid:1) ; ( x , x ) max n =5 = (cid:0) , (cid:1) ;Fig. 12: ( x , x ) max n =5 = (cid:0) , − (cid:1) ; ( x , x ) max n =5 = (cid:0) , − (cid:1) ;Fig. 12: ( x , x ) max n =5 = (0 , − x , x ) max n =5 = (1 , x , x ) max n =5 = (1 , − x , x ) max n =5 = (cid:0) , − (cid:1) . It will be interesting to observe where the experimental data at the LHC are placed: are they closerto regions with some correlation in rapidity space or in totally decorrelated maxima? Note that themaxima in the previous section and these coincide only for those minijets at the furthest distancefrom each other in rapidity with l = n and m = 1, i.e. ( x n , x ) = (1 , −
1) in this exploratory analysis.We can finish our discussion investigating Eq. (10). We summarize some results in the followingdescription of different figures where the red lines in each of them correspond to the values of the pairof rapidities for which R = 0 (the white regions correspond to sectors of very rapid growth of R ).The closer the experimental results lie to these regions, the more decorrelated the mini-jet productionwill be. We provide the analytic formula for these lines. We again only show the lower multiplicitiessince these will be statistically richer in the recent LHC data where the number of events decreasesrapidly with the multiplicity. In some cases there exist two branches for the lines of zeroes:Fig. 14: x = − x ) − x = √ √ − x ) − x = ± √ x (1+ x − x ) − x − x +53(1 − x ) ;Fig. 15: x = − x ) − x = / (1 − x ) / − x = ( ±√ √ x − x +2 x +1+2 ) (1 − x ) − x = − x (cid:16) h ( x ) + h ( x ) (cid:17) −
1; where h ( x ) = (cid:0) − (cid:1) / (cid:16) − x ± (cid:113) x ( x − − (cid:17) / .9 - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 3 - - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 4
Figure 6:
Left: Double differential cross section α Y d σ (3 , dy dy = ( x − x ) (1 − x ) = − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) (the re-lated by x → − x and x → − x distribution is α Y d σ (5 , dy dy = (1 + x ) ( x − x ) = − T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) ). Right:Double differential cross section α Y d σ (4 , dy dy = (1 − x ) ( x − x ) = T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) (the related by x → − x and x → − x is α Y d σ (5 , dy dy = (1 + x ) ( x − x ) = T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) ). In both x L = 2 y L /Y . - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 5 - - - - y Y y Y Multiplicity 5 + - Jet 2 vs Jet 4
Figure 7:
Left: Double differential cross section α Y d σ (5 , dy dy = ( x − x ) = T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) − T ( x ) T ( x )+ T ( x ) T ( x ) − T ( x ) T ( x ) . Right: Double differentialcross section α Y d σ (4 , dy dy = (1 + x ) (1 − x ) ( x − x ) = − T ( x ) T ( x ) + T ( x ) T ( x ) − T ( x ) T ( x ) − T ( x ) T ( x ) + 2 T ( x ) T ( x ) − T ( x ) T ( x ) − T ( x ) T ( x ) + T ( x ) T ( x ) . In both x L = 2 y L /Y . - - - y Y y Y Multiplicity 3 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 3 + - Jet 1 vs Jet 3
Figure 8:
Left: Product of cross sections α − (cid:0) Y (cid:1) − dσ (2)3+2 dy dσ (1)3+2 dy = (1 − x ) ( − x ) (the associated dis-tribution with x → − x is α − (cid:0) Y (cid:1) − dσ (3)3+2 dy dσ (2)3+2 dy = ( − x ) (1+ x ) ). Right: Product of cross sections α − (cid:0) Y (cid:1) − dσ (3)3+2 dy dσ (1)3+2 dy = (1 − x ) (1+ x ) . In both x L = 2 y L /Y . - - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 3
Figure 9:
Left: Product of cross sections α − (cid:0) Y (cid:1) − dσ (2)4+2 dy dσ (1)4+2 dy = (1 − x ) (1 − x ) (1+ x )12 (the related by x → − x and x → − x distribution is α − (cid:0) Y (cid:1) − dσ (4)4+2 dy dσ (3)4+2 dy = (1 − x )(1+ x ) (1+ x ) ). Right: Productof cross sections α − (cid:0) Y (cid:1) − dσ (3)4+2 dy dσ (1)4+2 dy = (1 − x )(1+ x ) (1 − x ) (the related by x → − x and x → − x distribution is α − (cid:0) Y (cid:1) − dσ (4)4+2 dy dσ (2)4+2 dy = (1+ x )(1 − x ) (1+ x ) ). In both x L = 2 y L /Y . - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 4 - - - - y Y y Y Multiplicity 4 + - Jet 2 vs Jet 3
Figure 10:
Left: Product of cross sections α − (cid:0) Y (cid:1) − dσ (4)4+2 dy dσ (1)4+2 dy = (1 − x ) (1+ x ) . Right: Product of crosssections α − (cid:0) Y (cid:1) − dσ (3)4+2 dy dσ ( m )4+2 dy m = (1+ x )(1 − x ) (1 − x )(1+ x ) . In both x L = 2 y L /Y . - - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 5 + - Jet 2 vs Jet 3
Figure 11:
Left: Product of cross sections α − (cid:0) Y (cid:1) − dσ (2)5+2 dy dσ (1)5+2 dy = (1 − x ) (1+ x )(1 − x ) (the related by x → − x and x → − x distribution is α − (cid:0) Y (cid:1) − dσ (5)5+2 dy dσ (4)5+2 dy = (1 − x )(1+ x ) (1+ x ) ). Right: Product ofcross sections α − (cid:0) Y (cid:1) − dσ (3)5+2 dy dσ (2)5+2 dy = (1+ x )(1 − x ) (1 − x ) (1+ x ) (the related by x → − x distribution is α − (cid:0) Y (cid:1) − dσ (4)5+2 dy dσ (3)5+2 dy = (1 − x ) (1+ x ) (1 − x )(1+ x ) ). In both x L = 2 y L /Y . - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 4 - - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 3
Figure 12:
Left: Product of cross sections α − (cid:0) Y (cid:1) − dσ (4)5+2 dy dσ (1)5+2 dy = (1 − x ) (1 − x )(1+ x ) (the related by x → − x and x → − x distribution is α − (cid:0) Y (cid:1) − dσ (5)5+2 dy dσ (2)5+2 dy = (1+ x )(1 − x ) (1+ x ) . Right: Productof cross sections α − (cid:0) Y (cid:1) − dσ (3)5+2 dy dσ (1)5+2 dy = (1 − x ) (1 − x ) (1+ x ) (the related by x → − x distribution is α − (cid:0) Y (cid:1) − dσ (5)5+2 dy dσ (3)5+2 dy = (1 − x ) (1+ x ) (1+ x ) ). In both x L = 2 y L /Y . - - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 5 - - - - y Y y Y Multiplicity 5 + - Jet 2 vs Jet 4
Figure 13:
Left: Product of cross sections α − (cid:0) Y (cid:1) − dσ (5)5+2 dy dσ (1)5+2 dy = (1 − x ) (1+ x ) . Right: Product of crosssections α − (cid:0) Y (cid:1) − dσ (4)5+2 dy dσ (2)5+2 dy = (1+ x )(1 − x ) (1 − x )(1+ x ) . In both x L = 2 y L /Y . - - - y Y y Y Multiplicity 3 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 3 + - Jet 1 vs Jet 3 - Figure 14:
Left: R ( x , x ) = σ d σ (2 , dy dy dσ (2)3+2 dy dσ (1)3+2 dy − (the associated distribution R ( x , x ) = σ d σ (3 , dy dy dσ (3)3+2 dy dσ (2)3+2 dy − is its mirror image wrt. the y = − y line). Right: R ( x , x ) = σ d σ (3 , dy dy dσ (3)3+2 dy dσ (1)3+2 dy − For figures 16 to 19 the analytic expressions are more complicated.This concludes our exploratory study of basic distributions which could be extracted from LHCdata in multi-jet events. The analysis is simple since it only takes into account the longitudinalsector of the phase space and does not introduce relevant dynamics in transverse momentum space.This nevertheless should be enough to capture the gross features of the proposed observables whichfocus on single and double rapidity correlations. These features should be global and independent ofthe selected p T in the twin-jets. In future works we plan to introduce the more refined transversespace information stemming from effective theories based on the concept of clusters in multi-particleproduction. In this context we will investigate whether the ideas here presented are connected topredictions derived from first principles in Yang-Mills theories such as the BFKL formalism eitheranalytically (see the next section for a treatment in the collinear region) or via the Monte Carlo code BFKLex [24–29].
In this section we would like to further motivate the use of the Chew-Pignotti model for the approx-imate description of the observables here discussed in a more modern language. It is not our targetto evaluate them with general purpose Monte Carlo event generators since these are not optimizedfor the particular kinematics under study. It is more natural to evaluate these quantities within theBFKL formalism which we will show that in its simplest form is equivalent to the Chew-Pignottimodel for rapidity correlations. After doing so, we will provide some kinematical cuts needed for afirst experimental investigation of our results. We leave for a future work a more complicated anddetailed study of the full BFKL dynamics in this context.At a hadronic collider running at center-of-mass energy √ s , in a leading logarithmic approximationwhere terms of the form ¯ α ns ln n s are resummed (with the QCD coupling ¯ α s = ¯ α s N c /π ), the differentialpartonic cross section for the production of two tagged jets in forward and backward directions withtransverse momentum (cid:126)p i =1 , and generated by gluons with longitudinal momentum fractions x i =1 , reads (with rapidity separation between the tagged jets Y ∼ ln x x s/ (cid:112) (cid:126)p (cid:126)p ) d ˆ σd (cid:126)p d (cid:126)p = π ¯ α s f ( (cid:126)p , (cid:126)p , Y ) (cid:126)p (cid:126)p . (12)14 - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 3 - Figure 15:
Left: R ( x , x ) = σ d σ (2 , dy dy dσ (2)4+2 dy dσ (1)4+2 dy − (the associated distribution R ( x , x ) = σ d σ (4 , dy dy dσ (4)4+2 dy dσ (3)4+2 dy − is its mirror image wrt. to the line y = − y ). Right: R ( x , x ) = σ d σ (3 , dy dy dσ (3)4+2 dy dσ (1)4+2 dy − (the associated distribution R ( x , x ) = σ d σ (4 , dy dy dσ (4)4+2 dy dσ (2)4+2 dy − is its mirror image wrt. the line y = − y ). - - - - y Y y Y Multiplicity 4 + - Jet 1 vs Jet 4 - - - - - y Y y Y Multiplicity 4 + - Jet 2 vs Jet 3
Figure 16:
Left: R ( x , x ) = σ d σ (4 , dy dy dσ (4)4+2 dy dσ (1)4+2 dy − . Right: R ( x , x ) = σ d σ (3 , dy dy dσ (3)4+2 dy dσ (2)4+2 dy − . - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 2 - - - - y Y y Y Multiplicity 5 + - Jet 2 vs Jet 3
Figure 17:
Left: R ( x , x ) = σ d σ (2 , dy dy dσ (2)5+2 dy dσ (1)5+2 dy − (the associated distribution R ( x , x ) = σ d σ (5 , dy dy dσ (5)5+2 dy dσ (4)5+2 dy − is its mirror image wrt. the line y = − y ). Right: R ( x , x ) = σ d σ (3 , dy dy dσ (3)5+2 dy dσ (2)5+2 dy − (the associated distribution R ( x , x ) = σ d σ (4 , dy dy dσ (4)5+2 dy dσ (3)5+2 dy − is its mirror image wrt. the line y = − y ). - - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 4 - - - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 3
Figure 18:
Left: R ( x , x ) = σ d σ (4 , dy dy dσ (4)5+2 dy dσ (1)5+2 dy − (the associated distribution R ( x , x ) = σ d σ (5 , dy dy dσ (5)5+2 dy dσ (2)5+2 dy − is its mirror image wrt. the line y = − y ). Right: R ( x , x ) = σ d σ (3 , dy dy dσ (3)5+2 dy dσ (1)5+2 dy − (the associated distribution R ( x , x ) = σ d σ (5 , dy dy dσ (5)5+2 dy dσ (3)5+2 dy − is its mirror image wrt. the line y = − y ). - - - y Y y Y Multiplicity 5 + - Jet 1 vs Jet 5 - - - - - y Y y Y Multiplicity 5 + - Jet 2 vs Jet 4 - Figure 19:
Left: R ( x , x ) = σ d σ (5 , dy dy dσ (5)5+2 dy dσ (1)5+2 dy − . Right: R ( x , x ) = σ d σ (4 , dy dy dσ (4)5+2 dy dσ (2)5+2 dy − . In the BFKL formalism, an effective field theory valid when Y is large, the gluon Green’s function f can be shown to follow, in a collinear approximation, the integral equation [CITATION NEEDEDPROBABLY] ∂f ( K , Q , Y ) ∂Y = δ ( K − Q )+ ¯ α s (cid:90) ∞ dq (cid:18) θ ( K − q ) K + θ ( q − K ) q + 4(ln 2 − δ ( q − K ) (cid:19) f ( q , Q , Y ) . (13)Its solution can be written in the iterative form f ( K , Q , Y ) = e − α s Y (cid:40) δ ( K − Q )+ ∞ (cid:88) N =1 (¯ α s Y ) N N ! (cid:34) N (cid:89) L =1 (cid:90) ∞ dx L (cid:18) θ ( x L − − x L ) x L − + θ ( x L − x L − ) x L (cid:19)(cid:35) δ ( x N − Q ) (cid:41) (14)with x = K . To connect with more standard representations we use δ ( K − Q ) = (cid:82) dγ πiQ (cid:16) K Q (cid:17) γ − and (cid:90) ∞ dx N (cid:18) θ ( x N − − x N ) x N − + θ ( x N − x N − ) x N (cid:19) (cid:16) x N K (cid:17) γ − = (cid:18) γ + 11 − γ (cid:19) (cid:16) x N − K (cid:17) γ − , (15)valid for 0 < γ <
1, to write (cid:34) N (cid:89) L =1 (cid:90) ∞ dx L (cid:18) θ ( x L − − x L ) x L − + θ ( x L − x L − ) x L (cid:19)(cid:35) (cid:16) x N K (cid:17) γ − = (cid:18) γ + 11 − γ (cid:19) N . (16)We then have f ( K , Q , Y ) = e − α s Y Q (cid:90) dγ πi (cid:18) K Q (cid:19) γ − ∞ (cid:88) N =0 (¯ α s Y ) N N ! (cid:18) γ + 11 − γ (cid:19) N = (cid:90) dγ πiQ (cid:18) K Q (cid:19) γ − e ¯ α s Y χ ( γ ) , (17)17here χ ( γ ) = 4(ln 2 −
1) + γ − + (1 − γ ) − .The partonic cross section is calculated after integrating over the phase space of the two taggedjets weighted by jet vertices Φ jet i = Φ (0)jet i + ¯ α s Φ (1)jet i + . . . , i.e.ˆ σ (cid:0) α s , Y , p , , l , (cid:1) = (cid:90) d (cid:126)q (cid:90) d (cid:126)q Φ jet (cid:0) (cid:126)q , p , l (cid:1) Φ jet (cid:0) (cid:126)q , p , l (cid:1) d ˆ σd (cid:126)q d (cid:126)q . (18)In the context of the observables here studied, at leading order, Φ (0)jet i (cid:0) (cid:126)q, p i , l i (cid:1) = θ (cid:0) l i − q (cid:1) θ (cid:0) q − p i (cid:1) is a product of two Heaviside step functions. Since the tagged jets are produced collinearly, thisparton level expression can be convoluted through the variables x i with standard parton distributionfunctions to generate hadron-level results. In order to reduce the influence from these pdfs and possibletheoretical uncertainties, it is convenient to work with normalised quantities in a preliminary analysis.At this point we can highlight that the BFKL formalism operates in the so-called multi-Reggekinematics for which Y (cid:29) p T emissions. If we keep | p i | ∼ | l i | this constrains the multiparticle phase space within theclass of multiperipheral final states here investigated. Therefore, for the class of two particle rapiditycorrelations studied in the previous sections, the predictions from BFKL should be very similar tothose in the Chew-Pignotti simple model.In Eq. (14) each N accounts for the emission of a new minijet and the three components of thekernel drive the virtuality of t -channel particles into infrared or ultraviolet regions with the deltafunction term being a non-Sudakov no-emission contribution. We can make use of Eq. (1) to write f ( K , Q , Y ) = ∞ (cid:88) N =0 ¯ α Ns (cid:90) Y N +1 (cid:89) i =1 dz i δ (cid:32) Y − N +1 (cid:88) s =1 z s (cid:33) ξ ( N ) ( K , Q ) (19)where ξ ( N ) ( K , Q ) = (cid:90) dγ πiQ (cid:18) K Q (cid:19) γ − χ N ( γ ) . (20)To single out particular emissions we apply the logic of Eqs. (2) and (7), i.e. df ( l ) N ( K , Q , Y, y l ) dy l = ¯ α Ns (cid:0) Y − y l (cid:1) N − l ( N − l )! (cid:0) y l + Y (cid:1) l − ( l − ξ ( N ) ( K , Q ) , (21) df ( l,m ) N ( K , Q , Y, y l , y m ) dy l dy m = ¯ α Ns (cid:0) Y − y l (cid:1) N − l ( N − l )! ( y l − y m ) l − m − ( l − m − (cid:0) y m + Y (cid:1) m − ( m − ξ ( N ) ( K , Q ) . (22)We can see that for normalized quantities the ξ ( N ) factor cancels out and we obtain the same predic-tions as for the Chew-Pignotti model. Needless writing that the full BFKL formalism carries moresubtle dependences in rapidity, transverse momentum and azimuthal angles which we will study indetail in future works. Nevertheless we find it interesting that in rapidity space it is not that far fromthe old multiperipheral cluster approaches.Let us conclude with a more concrete discussion of the range of parameters needed for a firstexperimental study of these effects in the 13 TeV data recorded by the LHC as well as the data fromthe previous run at 7 and 8 TeV. One should look at the relevant dijet experimental analyses for 7TeV data from both ATLAS and CMS [30–33]. There, more than one million of events per experimentcovering a rapidity span of more than 9 units was selected by tagging a dijet configuration in multi-jetfinal states after imposing a jet veto scale Q = 20 GeV for the crowd jets . Any other jet apartfrom the dijet system contributed to the jet multiplicity if it had a p T larger than Q . The outermost Here we follow the terminology we introduced in the present paper. p T as low as 35 GeV and probably lower, down to 20 GeV. Thismakes us confident that one will have good statistics when isolating events with N = 3, 4, 5 and 6jets in the final state (including the two most forward and most backward ones) although the drop inselected events for N = 6 will be considerable. With the advent of the 13 TeV data from the LHC at low luminosity numerous studies of multi-jetphysics can be performed. In the present work we suggest to investigate a particular set of eventswhere both a forward and a backward jet are clearly identified with similar transverse momenta, “twin-jets”. As the available phase space increases a “crowd” of low transverse energy mini-jet populatesthe gap between them. These are specific events within the class of Mueller-Navelet configurationsbut quite restricted in windows of p T in order to ensure that they belong to multiperipheral regionsof phase space with low momentum transfer (these have been detailed at the end of section 5). Wepropose their experimental study since in this set up it will be possible to distinctly identify featuresof different multi-particle production models such as those predicted by the BFKL formalism (adiscussion of its collinear limit has been introduced). We have performed a simple analysis based ona total decoupling of the longitudinal and transverse degrees of freedom and keeping the latter asa constant non-dynamical contribution. Within this context we have provided analytic predictionsfor single and double differential distributions in rapidity which can be compared with experimentalresults analyzed with equivalent fits to finite expansions on a basis of Chebyshev polynomials. Thiswill provide important information on the degree of correlations of this class of multi-particle finalstates in proton-proton interactions at very high center-of-mass energies.Future works will necessarily include the study of observables proving the QCD dynamics intransverse momentum and azimuthal angle space and their role in introducing short and, possibly,long range jet-jet correlations. Acknowledgements
This work has been supported by the Spanish Research Agency (Agencia Estatal de Investigaci´on)through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597 and the Spanish Govern-ment grant FPA2016-78022-P. It has also received funding from the European Union’s Horizon 2020 re-search and innovation programme under grant agreement No. 824093. The work of GC was supportedby the Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portugal) under project CERN/FIS-PAR/0024/2019and contract ‘Investigador FCT - Individual Call/03216/2017’.
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