PPomeron Physics at the LHC
Federico
Deganutti , ,? , David
Gordo Gomez , ,?? , Timothy
Raben ,??? , and Christophe
Royon ,???? Department of Physics and Astronomy, University of Kansas, 1082 Malott, 1251 Wescoe Hall Dr.,Lawrence, KS 66045-758 Dipartimento di Fisica ed Astronomia, Università di Firenze, Piazza di San Marco, 4, 50121 Firenze FI,Italia Instituto de Física Teórica UAM/CSIC, Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049Madrid, España & Universidad Autónoma de Madrid, E-28049 Madrid, España
Abstract.
We present current and ongoing research aimed at identifyingPomeron effects at the LHC in both the weak and strongly coupled regimesof QCD.
Hadron collisions in many kinematic regimes display Regge behavior where cross sectionsgrow as a power of the center of mass energy. Regge Theory grew out of pre-QCD S-matrixtheory. Amplitudes are seen as unitary, Lorentz invariant functions of analytic momenta. Poles in scattering amplitudes represent particle exchange or bound state production. Usinga partial wave analysis, the dominant contribution to simple amplitudes is the exchange ofan entire trajectory of particles: Pomeron exchange. Single Pomeron exchange is predictiveof a power-law rise in the total cross section σ tot ∼ s α − Since the beginning of Pomeronphysics, fits to p-p total cross section data have shown some power-law behavior. In theseproceedings we review progress in using Pomeron exchange to describe LHC data.
Weak Coupling Pomeron
Balitsky, Fadin, Kuraev, Lipatov (BFKL) asked, and provided the first important answersto, the question: what happens in the Regge limit of QCD? Large logarithms get in the way ? e-mail: [email protected] ?? e-mail: [email protected] ??? e-mail: [email protected] Speaker ???? e-mail: [email protected] The goal was to describe scattering experiments without a detailed description of an underlying micro-scopic theory. Generically amplitudes also have cuts corresponding to multiparticle production and non-linear interac-tions. The physics of cuts much less well understood and we omit any detailed discussion Asymptotically, as s → ∞ , cross-sections are likely subject to the Froissart bound indicating a maximalgrowth σ tot ∼ log ( s ). The contribution responsible for the empirically observed σ tot rise is referred to as the soft Pomeron .Although recent σ tot measurements show a deviation from a strict power-law, the soft Pomeron approach isstill used to fit data and for modeling.[1] a r X i v : . [ h e p - ph ] N ov q − k ′ q − kkk ′ = + . . . + | {z } G (0) ( k , q ) k q − k (a) (b) Figure 1. (a) Ladder diagram for the BFKL weak coupling Pomeron. (b) Cartoon of a 2-to-2scattering process using the AdS/CFT applicable for the BPST strong coupling Pomeron. of usual perturabtion theory; a resummation of terms α s log ( s ) to all orders is necessary.Scattering amplitudes are dominated by the t-channel exchanged of an infinite ladder ofreggized gluons (See Figure1(a)) leading to Regge behavior (power law growth of cross sectionswith energy). The BFKL equation, an integral equation for Green’s function in Mellin space,is used to described this exchange kernel. The solution can be written G ( k , k , q , Y ) = + i ∞ Z − i ∞ dω πi e Y ω f ω ( k , k , q ) → + i ∞ Z − i ∞ dω πi e Y ω X n ∈Z + i ∞ Z − i ∞ dγ π i E γ,n ( k ) E ∗ γ,n ( k ) ω − ¯ α s χ ( γ, n ) , (1) where in the leading log (LL) approximation: χ ( γ, n ) = 2 ψ (1) − ψ ( γ + | n | ) − ψ (1 − γ + | n | )and ω = α s N c π ln (2). Strong Coupling Pomeron
Pomeron phenomenology is applicable over a wide range of kinematics and, particularly if onefocuses on small momentum transfers, involves non-perturbative physics. The most promisingdescription of a unified soft/hard Pomeron framework comes from the AdS/CFT dualityconjecture. In this framework, a N = 4 susy Yang-Mills theory is dual to a string theory in ahigher dimensional curved space. The utility of such a duality is far beyond the scope here,but we mention a few pertinent properties: the duality is strong/weak so that a perturbative,weakly coupled string theroy calculation can give insights into strongly coupled Yang-Mills;the gluon sector of the N = 4 theory is similar to that of QCD; and the extra dimensions andnew fields become natural mechanisms for modeling the onset of confinement and saturation.Within the AdS/CFT duality, the Pomeron has been identified with the Regge trajectoryof the graviton . (Otherwise known as the BPST Pomeron[3]) Scattering amplitudes com-puted via Feynman diagram like approach (See Figure1(b) for a pictorial.) with the leadingRegge amplitudes written as a convolution of wavefunctions and Reggeon propagators overAdS space: A ∼ ψ ( z ) ψ ( z ) ∗ χ ( z, z , s, t ) ∗ ψ ( z ) ψ ( z ). Here s and t are Mandelstam likeinvariants and z is an AdS coordinate. Reggeon propagators are reminiscent of the weakcoupling partonic description χ R ∼ Z dj ( α ˆ s ) j (1 + cos( − iπj )) G j ( t, z, z ) (2)Operator dimensions of AdS Reggeons have an anomalous part and admit a non-trivial con-vergent expansion in terms of spin, coupling, and twist. These ∆ − J curves can be calculated The BPST Pomeron naturally connects a hard and soft regime and has diffusive behavior similar to theBFKL Pomeron. An initial description describing the first calculation of 1 / √ λ effects can be found in [2]. o high order using a mix of conformal, string, and integrability techniques. Minimizing thesecurves gives Reggeon intercepts.[4]Applications of the BPST Pomeron have included deep inelastic total cross section mea-surements from HERA[5–7], vector meson production[8] and diffractive Higgs production [9]at the LHC, to predict glueball masses [10], and to predict cross sections of non-diffractivecentral η production [11]. Here we review previous efforts, and describe our program in progress, for observing BFKLeffects at the current LHC collider kinematics . We are focused on exclusive and inclusivedijet production as in Figure 2. JetJethh f g/f f g/f gap (a) (b) Figure 2. (a) Mueller-Tang jet-gap-jet process. (b) Mueller-Navelet inclusive dijet process.
In the Mueller-Navelet (M-N)[12] jet process all the radiation is treated inclusively exceptfor the two jets farthest apart in rapidity. The link between this semi-inclusive process andthe purely elastic amplitude of the Pomeron exchange is provided by the optical theorem.Theoretically, this is one of the simplest jet processes sensitive to BFKL effects as the outermost dijet can be thought of coming from 2-to-2 forward ( t = 0) parton scattering. The BFKLequation greatly simplifies in this regime making both analytic and numerical computationsmuch more tractable; the BFKL gluon ladder is seen as an interference diagram with realemissions that contribute to the cross section instead of being part of the virtual correctionsto the 2 to 2 scattering amplitude.BFKL dynamics are predicted to manifest as an increase in the decorrelation of the mo-mentum of the tagged jets. The configuration of perfect correlation, valid at tree level, isincrisingly altered by the addition of any other emission. In the forward Regge limit, the ad-ditional real emissions are enhanced by the large log s . The larger is the rapidity separation be-tween the outer most tagged jets, the wider is the rapidity interval that can be spanned by theordered gluon emission, which generates the powers of logarithms: M-N jets with a large ra-pidity separation should be very sensitive to BFKL effects. The jet azimuthal (de)correlationcan be quantified considering the Fourier modes of the average cosine of the angular azimuthaldifference between the tagged jets C n /C m = h cos( n ( φ J − φ J − π )) i / h cos( m ( φ J − φ J − π )) i .Thanks to the high energy factorization, the coefficients C n are given by a convolution betweenthe probe dependent jet vertices and the universal gluon Green function: This kinematical regime is traditionally dominated by DGLAP evolution. m ( | k J | , | k J | , Y ) dY = Z dx dx f ( x ) f ( x ) dy J dy J δ ( y J − y J − Y ) dφ J dφ J × cos( m ( φ J − φ J − π )) Z d k k V ( k J , x J , k ) G ( k , k , ˆ s ) V ( k J , x J , k ) . (3)Here Y is the rapidity difference between the outgoing jets. The remaining finite part iscalled the jet vertex V ( k J , x J , k ). The jet vertices, as for the gluon Green function, dependon the approximation order and can be defined unambiguously in the BFKL approach. Theyhave been proven to be infrared safe up to the next-to-leading order.A test of the M-N BFKL predictions was carried out in 2013 by the CMScollaboration[13] . Events with at least two emerging jets with with transverse energy | k J | >
35 GeV and | y J | < . k T algorithm[15, 16] with adistance parameter R = 0 .
5. Results were compared to the BFKL analytical predictionsat NLL order and the predictions of several Monte Carlo generators like PYTHIA[19] andHERWIG++[20] which are based on the DGLAP evolution. y Δ 〉 ) φ Δ - π c o s ( 〈 DATAPYTHIA 6 Z2PYTHIA 8 4CHERWIG++ 2.5POWHEG+PYTHIA 6POWHEG+PYTHIA 8
CMS (7 TeV) -1
41 pb > 35 GeV, |y| < 4.7 T PMueller-Navelet dijets y Δ 〉 ) φ Δ - π c o s ( 〈 DATASHERPA 1.4NLL BFKLHEJ+ARIADNE
CMS (7 TeV) -1
41 pb > 35 GeV, |y| < 4.7 T PMueller-Navelet dijets y Δ 〉 )) φ Δ - π c o s ( ( 〈 DATAPYTHIA 6 Z2PYTHIA 8 4CHERWIG++ 2.5POWHEG+PYTHIA 6POWHEG+PYTHIA 8
CMS (7 TeV) -1
41 pb > 35 GeV, |y| < 4.7 T PMueller-Navelet dijets y Δ 〉 )) φ Δ - π c o s ( ( 〈 DATASHERPA 1.4NLL BFKLHEJ+ARIADNE
CMS (7 TeV) -1
41 pb > 35 GeV, |y| < 4.7 T PMueller-Navelet dijets y Δ 〉 )) φ Δ - π c o s ( ( 〈 DATAPYTHIA 6 Z2PYTHIA 8 4CHERWIG++ 2.5POWHEG+PYTHIA 6POWHEG+PYTHIA 8
CMS (7 TeV) -1
41 pb > 35 GeV, |y| < 4.7 T PMueller-Navelet dijets y Δ 〉 )) φ Δ - π c o s ( ( 〈 DATASHERPA 1.4NLL BFKLHEJ+ARIADNE
CMS (7 TeV) -1
41 pb > 35 GeV, |y| < 4.7 T PMueller-Navelet dijets
Figure 2: Left: Average h cos ( n ( p D f )) i ( n =
1, 2, 3) as a function of D y compared to LLDGLAP MC generators. In addition, the predictions of the NLO generator POWHEG interfacedwith the LL DGLAP generators
PYTHIA
PYTHIA
SHERPA with parton matrix elements matched to a LL DGLAPparton shower, to the LL BFKL inspired generator
HEJ with hadronisation by
ARIADNE , and toanalytical NLL BFKL calculations at the parton level (4.0 < D y < Figure 3.
From [13], average cos(∆ φ ), i.e. C /C , as a function of the rapidity difference Y compared to DGLAP MC generators on the left and to the analytic NLL BFKL calculations atpartonic level on the right. Similar analysis holds for most other ratios C m /C n . Concerning the BFKL predictions, few conclusions were drawn: the analytic NLL cal-culation predicts a decorrelation above the observed data for the three conformal momentsin the whole range of rapidities, except for the data points at Y ’
9. A better agreementwith data was observed for the ratios C n /C m ( m, n (cid:44)
0) along the whole rapidity range. TheDGLAP-based MC generator HERWIG++ reproduces the experimental data with satisfac-tory accuracy for all the observables.The analysis confirms that contamination from DGLAP evolution dijets affects substan-tially all the observables containing the total cross-section C and suggests that a description Sometimes in the literature ∆ y is used. Similar types of analyses have been done with dijet inclusive processes at electron-ion colliders. For arecent example see[14]. The NLO vertex found in the literature, and extensively used for theoretical analyses, is not in fullycompatible with the M-N prescription that is used in experimental analyses. The inconsistency can becorrected, and the difference can amount up to ∼ −
10% change. Although important, it does not qualitativelyalter the conclusions drawn in the analyses of CMS data. The corresponding paper is still unpublished[17],but an outline can be found in[18]. f such observables based on the BFKL approach alone is perhaps not feasible at the currentinitial energies; the kinematic domain explored in the CMS analysis seems to lie in a tran-sition region between the DGLAP and BFKL regimes for angular decorrelation. This opensthe path for alternative or complementary strategies to better disentangle BFKL effects ontop of the standard M-N angular decorrelation observables. (a) (b)
Figure 4.
Preliminary comparison of mini-jet p T and R y between BFKLex (red curves) and DGLAP-based PYTHIA8 (gold). PYTHIA results have been rescaled by one order of magnitude for ease ofcomparison. BFKLex data is from [21]. Since the geometry of the CMS detector limits the highest accessible rapidity, BFKL effectscannot be enhanced at LHC via increasing the center-of-mass energies. However, observablescould be more sensitive to the differences between DGLAP and BFKL evolutions. A promisingcandidate is to consider the same Mueller-Navelet definition with an additional requirementthat excludes events with an insufficient number of distinguished jets in between the taggedjets: counting mini-jet radiation. In fact, the BFKL ordered emission is expected to giverise to a larger mini-jet multiplicity compared to the collinear emissions re-summed in theDGLAP picture.In addition to multiplicity, other observables were recently proposed [22, 23] using thismini-jet radiation h p T i = 1 N X i | k i | , h θ i = 1 N X i θ i , hR y i = 1 N + 1 X y i y i − , (4)reflecting the average transverse momentum, emission angle, and rapidity ratio of the mini-jets respectively. Initial investigations showed that these observables could be simulated usingthe BFKLex MC to give non trivial predictions. In Figure 4, results of a comparison betweenDGLAP based PYTHIA8 and the BFKLex for the average transverse momenta and rapidityratio are plotted. These preliminary results[24] are extremely encouraging: the BFKL average p T has a very heavy tail and the BFKL R y is peaked differently from the DGLAP expectation.However, there are still challenges left to pursue. The data in Figure 4 has cuts forthe M-N outer jets similar to that of CMS, | p T | >
35 GeV. However, the mini-jets aresimulated with a rather modest | k i | > not fixed. This isprimarily an effect of not incorporating PDF effects in the MC; BFKLex only uses the BFKLkernel. This issue can be circumvented either by an including PDF effects in BFKLex or a V b G G q − k q − k ′ q − k ′ q − k k k ′ k k ′ + + NLOLOLL LL q − k q − k ′ q − k ′ q − k k k ′ k k ′ LONLOLL LL q − k q − k ′ q − k ′ q − k k k ′ k k ′ LOLONLL NLL q − k q − k ′ q − k ′ q − k k k ′ k k ′ Figure 5. (Top) Schematicrepresentation of the partonic elasticscattering via color-singlet exchange. G denotes the Green function and V the impact factors. (Bottom)Combinations of jet vertex andGreen’s function order thatcontribute to the full NLO M-Tprocess. normalizing the distributions in a region where there is a small overall rapidity gap and small p T difference between the outer most jets; a region where both DGLAP and BFKL effectsmight be suppressed and the predictions should agree. The consideration of new observables,like higher moment companions of R y might also alleviate these issues. The process of using large rapidity gaps to isolate and measure high energy asymptotic be-havior goes back to [25–27]. The properties of the BFKL hard Pomeron at finite momentumtransfer can be investigated at hadron colliders by looking for highly exclusive processes wherea dijet is separated by a large rapidity interval devoid of radiation [28]. These dijet events areknown as Mueller-Tang dijet events or jet-gap-jet events [29]. At the parton level the simplestconfiguration consists of 2-to-2 elastic parton scattering where, at large s, the non-forwardelastic amplitude is dominated by the exchange of a Pomeron. No real emission is allowed inthe internal rapidity region suppressing DGLAP evolution . Although this is a much cleanerexperimental signal of BFKL physics, handling the non-forward amplitude is a more difficulttask theoretically.An observable that gained large popularity is the differential cross section ratio of jet-gap-jet events to inclusive dijet events: R = dσ JGJ /dσ dijet . In the BFKL approach the jet-gap-jetpartonic cross section can be computed as d ˆ σdJ dJ d q = Z d k , d k , V a ( k , k , J , q ) G ( k , k , q , Y ) ×G ( k , k , q , Y ) V b ( k , k , J , q ) , (5)where J = { k J , x J } collects the variables that specify the jets. It is represented schematicallyin Fig. 5. The non-forward gluon Green’s function and the jet vertices depend on theapproximation order .To identify M-T events, the two hardest jets resulting from the color-singlet exchange areselected which: are strongly correlated in their azimuthal angular separation; are balanced This can be seen in Figures7(a) and 8(a) in the large excess of BFKL events at zero multiplicity. The observable R contains the gap survival probability S . This has an intrinsic non-perturbative nature[30–34] In most applications it has been assumed to be an empirically determined constant depending only onthe center of mass energy, although recent results have shown it should have some kinematic dependence[35,36]. If the jet vertices are kept at leading order the general expression can be greatly simplified. This fact hasbeen extensively used in the phenomenological analyses. igure 6.
Comparisons between the D0measurements of the jet-gap-jet event ratio usingthe NLL-BFKL kernel (solid line), LL-BFKL(dashed line) predictions with respect to the 2-to-2NLO DGLAP prediction. Taken from [37]. TheNLL calculation is in fair agreement with the datawhile the LL one leads to a worse description. in transverse momenta; and lie on opposite hemispheres of the central detector ( η · η < E > | η | < √ s = 1 . . The rapidity gap condition was applied only on the centralrapidity region | η | < p T > . p T and the rapidity difference ∆ η . Even in this regime, the BFKLcan be used to compute the partonic elastic amplitude. The implementation in a MonteCarlo is necessary in order to make a comparison with data; cross section measurementsare sensitive to the jet size and the observed rapidity separation between the leading jets issmaller due to the soft radiation from the edge of the jets. Underlying event effects, whichare estimated to be small, with the latest MC tuning should also be incorporated in thesimulation. A phenomenological study was done of data released by the D0 Collaboration[37]to the BFKL prediction by implementing the NLL BFKL kernel into HERWIG6.5. (SeeFigure6). Importantly it was found that improving the BFKL kernel to NLL order andincluding a large number of conformal spin contributions are both necessary to obtain goodagreement with data. In the CMS analysis summary, the results are compared only with theLL BFKL predictions (See Fig. 7(b)). These plots hint the need of higher-order corrections.The hard pomeron exchange embedded into HERWIG6.5 lacks the high order jet vertexcorrection to complete the NLL order. Nonetheless, it is expected that once the normalizationis fixed for a given data set the NLL BFKL is able to describe reasonably well the Tevatrondata.The charged multiplicity distribution measured by CMS in Fig. 7(a) is well described byDGLAP models (PYTHIA6) except at zero multiplicity, where we expect to see the jet-gap- This is the only experimental analysis coming from the LHC to date, though there is an ongoing 13 TeVanalysis according to private communications with members of the CMS collaboration. The rescattering of proton remnants can destroy the gap resulting in a suppression factor that can affectat least the overall normalization.a) (b)
Figure 7.
From [40]: (a) Number of charged particle tracks N Tracks for 40 GeV < p subleadT <
60 GeVwith | η , | > . η · η < | η | < f CSE dijet events as afunction of the subleading p T measured for √ s =7 TeV, compared to the LL BFKL predictions for N Tracks = 0. A large numeric and qualitative disagreement between the two can be seen. jet events. The excess is consistent with the prediction of the HERWIG6.5 event generatorusing the color-singlet exchange process, even if the LL approximation BFKL kernel is used.There is a hint that the Mueller-Tang jets is an excellent venue to extract the signal of aBFKL Pomeron, even if the event generation was done with the LL approximation kerneland the leading conformal spin. Refining the theoretical prediction up to the full NLL BFKLorder is an essential step toward this goal to show that the hard color-singlet exchange givesthe correct differential distributions.
Until now the description of M-T jets has focused on incorporating the NLL BFKL kernel,considering higher conformal spin corrections, and understanding soft events that can affectthe rapidity gap signature. [37, 41–43]. The path forward is clear: add the NLO jet vertex to the NLL BFKL kernel and perform a full NLO analysis of LHC data. Still, the full NLOcalculation is numerically very non-trivial. The convolutions over the transverse momentain Eqn. 5 can be computed analytically if the vertices are kept at the LL approximation.At NLO, they have to be solved numerically utilizing Monte Carlo integration techniques.Similarly to the M-N process, the corrections are expected to be large and important to ac-curately predict BFKL physics, though the calculation is absolutely necessary to verify this.In [37], to speed-up computation time, the amplitude is fit to an ad-hoc parameterizationand fed into HERWIG6.5 in place of standard QCD 2-to-2 process. More recently a varietyof parameterizations was tested[48] and preliminary results show that the best fit parameter- A group of Uppsala University has studied the jet-gap-jet process since the time of the Tevatron [44, 45].They have developed the HARDCOL package, a modified version of the PYTHIA6 event generator, wherethey embedded the solution of the non-forward LL BFKL equation for the jet vertices and kernel. Thisimplementation considers also the multi parton-parton interactions with the latest tuning in PYTHIA6 basedon LHC data at 7 TeV. In addition, they used the Soft Color Interaction model for color rearrangements in thefinal state through soft gluon exchanges, since such rearrangements can have large effects on rapidity gaps.This approach is complimentary to that proposed here. Recently the NLO jet vertices were calculated[46, 47] using Lipatov’s effective action. Although thevertices were shown to be finite, there is a careful cancellation of soft and collinear divergences between thereal and virtual corrections. In addition, the jet definitions involve complicated non-analytic pieces to preventgap contamination. Both of these require careful numerical treatments. zation can be physically motivated by considering asymptotics of the NLL kernel. Figure 8was generated using this parameterization. Jets are reconstructed using the anti- kt algorithmwith a distance parameter of R = 0 . | < 1 h Particle multiplicity in | E v en t s Entries 870
BFKL NLL (Kernel only), HERWIG6QCD2to2, PYTHIA8 ) -1 = 13 TeV (400 nb s (a) h D E v en t s / . ) -1 = 13 TeV (400 nb s BFKL@NLL, HERWIG6 (b)
Figure 8. (a) Particle multiplicity in the fixed rapidity region | η | <
1. Stable particles pass theselection cut p T > The BFKL Pomeron, BPST Pomeron, and QCD as extremely short distances exhibit con-formal symmetry . However, conformal symmetry has historically eschewed a description ofscattering process because of the inability to define asymptotic states. Using the AdS/CFT,it was recently shown[49, 50] that effects due to conformal properties can be exhibited inLHC scattering. The central idea, extended in these works, is that instead of consideringthe scattering of asymptotic states, the emphasis should be on the flow of IR safe quantities.In addition, by utilizing generalized optical theorems, cross sections can be expressed as adiscontinuity over the appropriate amplitude. For example, the differential cross section for It is important to note that they are different symmetries. / 1 GeV T p1 10 ) - ( G e V T dp h N / d ) d e v t N T p p / ( - - - - - - - - - - = 5.02 TeV NN sALICE Data at |<0.3 h | 4 · <-0.3 h -0.8< 16 · <-0.8 h -1.3< Figure 9.
Fits of the ansatz, Eq.(6), to the ALICE √ s NN = 5 . able 1. Fitting parameters for holographic description of inclusive central production in p-Xscattering.
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