Jet properties at high-multiplicity
IIPPP/14/104, DCPT/14/208, MCnet-14-24
Jet properties at high-multiplicity
Erik Gerwick ∗ and Peter Schichtel
2, 3, † II. Physikalisches Institut, Universit¨at G¨ottingen, Germany Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Germany, and Institute for Particle Physics Phenomenology, Durham University, UK
We investigate the behaviour of jets at high-multiplicity using analytic techniques. We considerin detail the rates, areas and the intermediate splitting scales as a function of the number of jets. Ineach case, we are able to characterise a general scaling behaviour characteristic for QCD processes,which we compare with results from the parton shower. The study of these observables potentiallyoffers a very general handle in the difficult to describe regime of high-jet multiplicity.
Contents
I. Introduction II. Jet Ratios from the Gen- k T generating functional III. Distribution of k t splitting scales IV. Conclusions A. Exponentiated form of the evolution equation B. Closed solution in the staircase limit References ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - ph ] D ec I. INTRODUCTION
Quarks and gluons produced in hard scattering reactions are measured experimentally as QCD jets. Connectinghard scattering to the hadronic final state is an extended regime of parton evolution, where a large number ofperturbatively well defined splittings produce particles mostly in the direction of hard partons. In some casesthese also become jets so that the final state contains an appreciable number of radiated jets.The theoretical description of radiated jets is often done via parton shower simulation [1–3]. Though verysuccessful in some regards, there are two downsides to this. First, the parton shower is limited in formal accuracy,and second, features of the resulting distributions may be opaque from only the individual components. Whilewe will say very little about the first point, fortunately, there is an analytic formalism, the generating functional,which allows us to address the second.For jets defined using modern jet-finding algorithms, the generating functional was constructed first for theDurham algorithm [4]. Jet rates in this algorithm are defined in terms of a single dimensionless parameter y cut . However, jets at hadron colliders are typically clustered using the generalised- k t class of jet algorithms,for example the k t [5, 6], anti- k t [7] or Cambridge-Aachen [8, 9] algorithm. Here jets are defined in terms of aminimum scale E R and radius R . A generating functional valid for these algorithms, and resumming logarithmsof the type α S log( E/E R ) log (1 /R ) was introduced in Ref. [10].A useful feature of the generating functional formalism is that the multiplicity distribution can be understoodto all-orders in perturbation theory. An example is the analytic solution for the average jet multiplicity [10, 11].Similarly, the distribution differentially in the jet multiplicity can also be computed, and in the case of theDurham algorithm was in Ref. [12] for particular kinematic limits. It was found that jet rates tend to followone of two scaling patterns [13], which are most easily understood by considering the ratio of successive jetrates, σ n +1 /σ n . In the large logarithmic limit log(1 /y cut ) (cid:29)
1, the distribution is that of a Poisson processand follows
Poisson scaling , where σ n +1 /σ n ∼ / ( n + 1). On the other hand, in the limit log(1 /y cut ) > α S log(1 /y cut ) (cid:28)
1, the distribution becomes geometric σ n +1 /σ n ∼ constant, which was deemed staircasescaling . While some insights on scaling were given in Ref. [10] for the Gen- k t class of jet algorithms, a completederivation of the multiplicity distributions was not.The motivation for this paper is to extend (and generalise) the notion of scaling in the Gen- k t class ofjet algorithms via the Gen- k t generating functional. While the idealised scaling patterns can be worked outanalytically, the distributions coming from parton shower simulation rarely follow these patterns exactly. Aspointed out in Ref. [10], the discrepancy partially originates from effects which are not included in the generatingfunctional formalism, for example kinematics and finite area effects, which disappear only in the exact limits.For realistic jet radii ( R = 0 . − . k t splitting scale as a function of the multiplicity. After computing someanalytic results, we are able to frame the study of this observable in the context of idealised scaling patterns,one representing a perfect Poisson process, and the other for an idealised non-Abelian splitting history.Our motivation for understanding multiplicity distributions which undergo scaling is that when these propertiesare sufficiently generic, they provide useful handles in difficult QCD environments. For example, while it wouldbe very challenging to predict exact properties of a 20 (sub)-jet final state, it is much more feasible to use theshape of first principle distributions in multiplicity (comparing to say all 10 - 19 jet events) in order to constrainwhether 20 (sub)-jet events are consistent with the QCD background as a whole.A significant amount of recent progress has come from applying analytic techniques to QCD intensive observ-ables, particularly in the context of sub-jet studies [14, 15]. In the direction of jet multiplicities and a numberof other (sub)-jet properties, very recently logarithms of the type α S log(1 /R ) were resummed in Ref. [16],which are not considered to all-orders in the present work. Further studies on the impact of jet algorithms onresummation, and especially the appearance of logarithms in R are given in Refs. [17–19].This paper is arranged as follows. We start off in sec.II with a brief review of the Gen- k t generating functionaland proceed to compute the multiplicity distribution. We then attempt to quantify area effects and perform adedicated comparison to Montecarlo. In sec. III we explore the average splitting scales for different multiplic-ities and splitting histories. We offer some discussion on ideas in sec. IV for the applicability of this work inphenomenological studies, which are saved for future work. In the appendix we provide more details on the ratecalculations performed in this work. II. JET RATIOS FROM THE GEN- k T GENERATING FUNCTIONAL
We start here by reviewing the Gen- k t generating functional. A more in depth description is found in Ref. [10].Our starting point is the fully exponentiated generating functional (see app. A) for a parton of flavour i . Interms of the opening angle ξ = 1 − cos θ between the emitter and emitted parton, and the energy ratio of thescale evolution e = E/E R , we haveΦ i ( e, ξ ) = u exp ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) /e dz α s ( ze, ξ (cid:48) )2 π (cid:88) j,k P i → jk ( z ) (cid:18) Φ j ( e, ξ (cid:48) )Φ k ( E ( z ) , ξ (cid:48) )Φ i ( e, ξ (cid:48) ) − (cid:19) , (1)where i, j, l ∈ { q, ¯ q, g } . The function E ( z ) = ze except for g → q ¯ q where it is e . The one loop running couplinggiven by α S ( ze, ξ (cid:48) ) = πb log z e E R ξ (cid:48) Λ , (2)is defined in terms of the coupling at the hard scale. The − i ( e, ξ ) = exp − ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) /e dz α s ( ze, ξ (cid:48) )2 π (cid:88) j,k P i → jk ( z ) , (3)interpreted as the no-emission probability between scales E and E R and angular distances from the hemisphereboundary to the cut-off set by ξ R = 1 − cos R , defined in the frame of the emitter. For jet production in theentire phase space one takes ξ = 1 − cos ( π/
2) = 1. If the observable is related to sub-jets inside of a larger jetof radius R L , the correct upper boundary is ξ = 1 − cos ( R L ).Finally, the n -th jet rate is obtained by differentiating Eq. (1) with respect to the parameter uσ n = σ n ! d n du n Φ i ( e, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) u =0 . (4)A few notes are in order regarding the approximation of the following calculations. Analytic results are oftenquoted at finite order and at fixed coupling for clarity, while the full results are numerically evaluated in theplots where stated. In general, we will use the small R approximation so that 1 − cos R ≈ R / α S log( e ) log( ξ/ξ R ) in the exponent. A. QCD scaling limits
Scaling patterns in jet-rates correspond to two idealised statistical distributions. The first is Poisson scalingwhere σ n +1 σ n ≡ R n +1 n = ¯ nn + 1 , (5)with ¯ n the average number of jets. Eq. (5) is characteristic for the large logarithmically dominated regime ofQCD. Staircase scaling on the other hand corresponds to the geometric (or fractal) regime of QCD radiation R n +1 n = R (6)where R is a constant. The extent to which jet distributions at ATLAS follow these patterns was studiedexperimentally in Ref. [20]. A correspondent pheno study can be found in [21]. Using high-precision multi-jetNLO calculations this type of behaviour was investigated in [22, 23]. In the context of BSM searches, Ref. [24]studied staircase scaling for SM backgrounds opposed to new physics decay jets, while Ref. [25] probed the extentof a staircase like distribution in sub-jet multiplicities.We will now derive Eq. (5) and Eq. (6) in the Gen- k t generating functional formalism. Poisson scaling
In the double-logarithmically dominated regime of QCD we expect the rates to become a Poisson process [12,13]. For the Durham algorithm this is achieved with a small resolution parameter y cut . The generalized- k T version depends in fact on two scale choices. The spacial jet resolution ξ R and the allowed energy range definedby E R (= p min T ). In the limit 1 ≥ ξ (cid:29) ξ R the integral is dominated by the ξ (cid:48) ≈ ξ R region. Thus we findΦ i ( e, ξ ) = u exp ( u − ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) /e dz α s ( ze, ξ (cid:48) )2 π (cid:88) j,k P i → jk ( z ) , (7)which is known to produce Poisson scaling. The limit 1 /e → /z which means that the z ≈ g → q ¯ q splitting drops out as its splitting kernel is not divergent but goes tozero in this limit. Note that starting with several hard partons we similarly produce a Poisson process [10, 13].However, we expect that the argument breaks down at some multiplicity around n ≈ ¯ n so that R n +1 n <
1, where¯ n is the rate parameter of the Poisson process. Staircase scaling
From the generating functional in the Durham algorithm we know that the staircase limit is the one oppositeto the large double-log limit. Formally, this regime exists when α S log( e ) log(1 /ξ R ) (cid:28)
1. Thus we study Eq. (1)in the e → /ξ R ) (cid:29)
1. We also focus on the pure Yang-Mills case for simplicity, though the argumentsare more general. We Taylor expand the integrand of the generating functional around z ≈ z < g ( e, ξ ) = u exp ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) /e dz α s ( z, ξ (cid:48) )2 π P g → gg ( z ) (cid:32) Φ g ( e, ξ (cid:48) ) + ∞ (cid:88) n =1 ( e ( z − n n ! d n Φ g ( e, ξ (cid:48) ) de n − (cid:33) . (8)Taking only the leading behavior of the n = 1 term into account we are able to find a closed solution for Φ g (seeapp. B) Φ g ( e, ξ ) = 11 + (1 − u ) u ∆ g ( e,ξ ) − ( u − χ ( e, ξ ) ≈
11 + (1 − u ) u ∆ g ( e,ξ ) + ( u − χ ( e, ξ )(1 + (1 − u ) u ∆ g ( e,ξ ) ) . (9)The function χ ( e, ξ ) is given in the app. B. By taking successive u derivatives we find that the jet ratios are R n +1 n = (1 − ∆ g ( e, ξ )) (cid:34) (cid:18) (1 − ∆ g ( e, ξ )) χ ( e, ξ )∆ g ( e, ξ ) − g ( e, ξ ) − n + 1) (cid:19) − (cid:35) . (10)We can check the large n limit of this formula by taking the resolved limit α S L (cid:28) /n ). This is in agreement with the fitted form of the resolved coefficients as derivedin Ref. [26].The result in Eq. (10) is very similar to the one found using the Durham algorithm in [12]. At high multiplicitythe ratios converge to constant staircase scaling. However, we find a very interesting additional feature: astaircase breaking term indicating that pure staircase scaling is an asymptotic feature in n . The breaking termenhances the low multiplicity ratios with respect to pure staircase scaling. Note that, as mentioned above,formally we need small ξ R . For finite values we expect ξ R dependent deviations from Eq. (10). B. Inclusion of Area effects
Naive Phase space effects
Since we will be studying the high multiplicity behaviour in detail, an additional effect we would like to includeis the simple depletion of geometric area as the jet multiplicity increases. The assumptions we make are that the
Figure 1: Left: The harder jet (red) defines a disk like object, while the softer jet (blue) is partially coverd. Thisdefines Eq. (15). Middle: Flower like configuration of collinear jets as included in our analytic approach. Right: Non-trivial overlap configuration demonstrating that very small jet areas are possible (green). geometric area of each jet is approximated by πR , and that the rates decrease proportional to the availablearea uniformly. This uniquely amounts to an overall suppression factor at the level of the ratios φ ps ≡ φ ( n + 1) φ ( n ) = 1 − ( n + 1) R / − nR / R n +1 n → φ ps R n +1 n . At some n the numerator will reach 0 signifyingthat naively, the entire phase is saturated, and the ratios go to 0. Although we will later see that this is aninsufficient approximation it is very useful for some order of magnitude estimates. We investigate the exactphase space effects in detail in the next paragraphs. Evolution equation for jet areas
In this section, we study how the distribution of jet areas ρ ( A ) evolves with higher multiplicity. Jet areas inthe generalised- k t class of jet algorithms was studied first in Ref. [27], with many techniques implemented inRef [28]. It is a well known feature of the anti- k T algorithm that it produces circular jets [7]. Furthermore, onlywith this algorithm is the area observable IR safe. We therefore orient this section towards specifically anti- k T .Due to the collinear nature of QCD jets quite often overlap, as depict schematically in the left panel of Fig. 1.This fact causes the jet areas to follow a non-trivial distribution in multiplcitiy. We assert that for a fixed numberof jets n the distribution of areas can be parameterised as ρ n ( A ) = ¯ ρ n δ ( A − ¯ A full ) + (1 − ¯ ρ n ) ρ overlap n ( A ) , (12)where ¯ A full = 2 π (1 − cos( R )) ≈ πR is just the geometric size of a jet of radius R projected on the sphere and¯ ρ n denotes how many full area jets we expect, which encodes one part of the non-trivial evolution. The othernon-trivial part is described by ρ overlap n ( A ), which is a result of how the jets overlap.What can we learn about these two quantities? Let us start with the simplest case n = 2, where ¯ ρ = 1.Therefore, we find trivially ρ ( A ) = δ ( A − ¯ A full ) . (13)From the collinear approximation we know that jets are distributed due to dξ/ξ . Assuming fixed coupling andnoting that the full area jet, which is already present, yields a lower bound for the integration we find¯ ρ = log (4) − (cid:0) csc ( R ) (cid:1) log (1 − cos ( R )) . (14)To find ρ overlap n ( A ) we need to study how the geometrical overlap of two jets works. For n = 3 we can constructthat the non-covered part of the area of a collinear jet which is partially covered by a full area jet is A not-covered ( ξ ) = R (cid:32)(cid:114) ξ − ξ R + 2 R arcsin (cid:32)(cid:114) ξ R (cid:33)(cid:33) ≈ πR − (cid:114) (cid:16) R − R (cid:112) ξ + ξ (cid:17) . (15)Noting that Eq. (15) implies a lower bound of¯ A min3 = 16 (cid:16) √ π (cid:17) R (16)for the minimal jet size and using again that jets are distributed due to dξ/ξ we compute ρ overlap3 ( A ) = 6 Θ (cid:0)(cid:0) ¯ A full − A (cid:1) (cid:0) A − ¯ A min3 (cid:1)(cid:1) A − √ π ) R + 4 √ R (cid:113) − A + (cid:0) √ π (cid:1) R × (cid:16) π √ (cid:17) − (cid:18) √ √ − π (cid:19) . (17)In the collinear region this equation is exact. One might be concerned that the transition to the δ -functionfor full area jets is not smooth. Indeed, if we expand Eq. (15) around ξ = (2 R ) / R / /
2. This causes the measure of the inverse function to diverge. The result isa divergent, yet integrable, behaviour of ρ overlap3 ( A ≈ πR ). We do not try to merge the two behaviours here,because this region is shielded by finite resolution effects as we will see in the following section.For higher jet multiplicities the situation gets more complicated. Schematically moving from n to n + 1 jetsis described by ρ n +1 ( A ) = 1 n + 1 (cid:0) n ρ n ( A ) + ¯ ρ n +1 , δ (cid:0) A − ¯ A full (cid:1) + (1 − ¯ ρ n +1 , ) ρ n +1 , ( A ) (cid:1) . (18)Here 1 − ¯ ρ n +1 , is how likely it is to add another covered jet. With ρ n +1 , ( A ) we describe how the area of this jetis distributed. For lower multiplicities we might hope to describe both with their ρ equivalents. Deviations fromthis assumption come from configurations where the additional jet overlaps with more than one of the previousones, which is a configuration we cannot easily describe with naive geometrical considerations. Of course, froma certain multiplicity on these configurations will dominate. One of the features that result from this fact is thatwe will not have a sharp drop at ¯ A min3 anymore, but a smoother distribution allowing all possible values for A .In order to estimate at which value n for the jet multiplicity this will become significant, let us perform somecounting gymnastics for R = 0 .
5. Some leading jets could cover three to four other jets without them touchingeach other. Such a configuration would look like a kids version of a flower, see center panel of Fig. 1. For tighterconfigurations we are forced to produce non-trivial overlaps as shown in the right panel of Fig. 1. Therefore, weshould see a non-trivial structure from starting at 5 or 6 jets. However, choosing this exact configuration is ratherunlikely, if we remember that there is a chance O (¯ ρ ) to produce full area jets instead. From the naive phasespace considerations, see Eq. (11), we know that around n = 15 the non-overlap picture must break down. Thismeans that between n = 6 and n = 15 the non-trivial structure becomes significant. The simplest guess would bethat we are able to describe jet area distributions with successive usage of ρ up to n ≈
10. Using Eq. (18) we seethat we have (cid:104) A n (cid:105) < ¯ A full . This leads to more jets then naively expected. However, generalizing configurationslike in the right panel of Fig. 1 it is clear that this still under-estimates the maximum number of possible jets.The end point of the spectrum is non-trivial and we will study it in more detail in the next section. Jet area distribution from MC
In this paragraph we would like to compare our analytic considerations with a parton shower simulation(specifically the CS shower in SHERPA [29, 30]) in order to assess non-trivial behaviour at high multiplicities.For this purpose it is instructive to recall how jet areas are measured. In the Fastjet implementation [27, 28] jetareas can be computed in one of three ways, active area, Passive areas and Veronoi areas. We use the activearea option of Fastjet, but implement spherical coordinates. The active area option divides the ( η, φ )-plane intocells of fixed size and places a ghost particle (almost vanishing p T ) randomly in each cell. If the ghost happensto end up in a certain jet, we count the cell’s size towards that jet’s active area. A0.5 0.6 0.7 0.8 ( A ) n r analyticanalyticanalyticn = 3n = 5n = 7jet area distribution R = 0.5, e = 10 fl collinear enhancement A0.5 0.6 0.7 0.8 ( A ) n r analyticanalyticanalyticn = 4n = 6n = 8jet area distribution R = 0.5, e = 10 fl collinear enhancement n10 20 30 40 50 60 70 ( n ) f f naive R = 0.3 f naive phase space Figure 2: Left and middle panel: Comparison of ρ n ( A ) from Eq. (12) (dots) with MC data for R = 0 . n = 3 upto n = 8 (shaded)xs. Right panel: Comparison of the actual phase space computed via (cid:104) A ( n ) (cid:105) (dots) with the naiveformula Eq. (11) (dashed) and a polynomial of third order (solid) for R = 0 . R = 0 . We study jets of size R = 0 . R = 0 .
3. We will also consider jets of size R = 0 . n ≈ A full . Using a random sample of ghostsand jet midpoints we expect that the area is normal distributed around A = 0 .
77 with a width of ∆ A = 0 .
02 for R = 0 . A = 0 .
28 with a width of ∆ A = 0 .
016 for R = 0 . A is driven by the cellsize and the actual jet radius.We compare these the analytic formula from the previous section with MC data, using the numbers in theprevious paragraph. For the average jet size and width we find good agreement. However, the actual width isslightly larger due to spherical geometry. In Fig. 2 we show the non-trivial area distributions for the exclusive n = 3 up to n = 8 cases for R = 0 .
5. We find good agreement for the three jet case with Eq. (17). For highermultiplicities we fix the overall normalisation of Eq. (18) to produce the same maximum height as the peak at¯ A full in the MC ∗ . We observe that we describe the area overlap distribution accurately for low multiplicities. Asexpected, for higher multiplicities n ≈ O (10) our description breaks down. In the right panel of Fig. 2 we showthe phase space for all multiplicities for R = 0 . R = 0 . φ ( n ) = 4 π − (cid:104) A ( n ) (cid:105) . The true phase space follows a polynomial ofthird order, where the linear part is driven by Eq. (17). The higher order terms encode the non-trivial overlapconfigurations not accessable with our ansatz. C. The full picture
Now we would like to put all the different parts together, and see to what extent we can understand theeffects driving the jet ratio distribution. In order to maximise statistics we choose a very large energy hierarchy E = 10 GeV and E R = 1GeV, with jets clustered in the anti- k t jet algorithm.We expect low multiplicities to be described by a Poisson process, defined by ¯ n = | log(∆( e, ξ )) | [12]. Inaddition the average number of jets is rather sensitive to the chosen coupling at the hard scale, which we takeas α S ( E ) = 0 . R effects are expected from the generating functional as well as the fact that very low jet multiplicitiesare matched to the matrix element. At multiplicities n (cid:29) ¯ n we have Eq. (10). In between, there is a somewhatawkward intermediate regime which we cannot say much about. However, for realistic jet radii O (0 .
5) thisregime is very small. For high multiplicities the pure staircase scaling is suppressed by phase space effects drivenby depletion of available area as shown in Fig. 2. Note, however, that for finite R we expect additional effects,which are not included in our approach. These should vanish as R → R = 0 . R = 0 . R = 0 . e = 10 in Fig. 3. The green line corresponds to the Poisson hypothesis, while the yellow dashed line ∗ This is neccessary because we neglect any possibility of non-trivial overlap for our analytic ansatz. This directly leads to a differentnormalization for the two quantities we wish to compare, which we compensate with this prescription. nn+11/0 6/5 11/10 16/15 21/20 nn + R R = 0.5, e = 10naive phase spacePoisson = 5.2n phase-space · staircase = 0.747, B = 4.8 R = -0.0177dn dR nn+11/0 16/15 31/30 46/45 nn + R R = 0.3, e = 10naive phase spacePoisson = 6.13n phase-space · staircase = 0.835, B = 7.8 R = -0.0039dn dR nn+11/0 31/30 61/60 91/90 121/120 151/150 nn + R R = 0.1, e = 10naive phase spacePoisson = 8.02n phase-space · staircase = 0.927, B = 8.6 R = -0.00035dn dR Figure 3: Exclusive jet cross section ratios for R = 0 . R = 0 . R = 0 . dR /dn ( n + 1) is fitted to quantify thedeviation from Eq. (10). In addition we show the naive phase space assumption (dashed yellow) and the Poisson scalinghypothesis (green). specifies Eq. (11). The red line is charecterized by (see Eq. (10)), R n +1 n = (cid:18) R (cid:20) B + ( n + 1) (cid:21) + dR dn ( n + 1) (cid:19) × φ ( n + 1) φ ( n ) , (19)where the numerical values for R , B , and dR /dn are given in the plots. We take the phase space factor fromthe polynomial in the previous section. Note, while there is a deeper connection between R and B , as well asthere exact position Eq. (19) that is not true for the term dR /dn . This term is purely phenomenological andwe introduce it to describe an additional small tilt we observe for jet cross section ratios. Therefor it quantifiesthe discrepancy from our staircase scaling formula. Recall that we formally need small R for the derivationof Eq. (10). In Tab. I we show that the fitted deviation dR /dn is small and indeed vanishes as R → III. DISTRIBUTION OF k t SPLITTING SCALES
Intermediate splitting scales in multi-jet events are useful primarily due to the close correspondence withparton shower splitting variables. They have been measured extensively in collider experiments, in e + e − [31]and more recently at the LHC in the context of ME-PS matching [32]. As our emphasis in this work is scalingfeatures especially at high-multiplicity, we explore in this section the distribution of splitting scales as a function ofmultiplicity very much in analogy to the jet rates. The hope is that the resulting distribution could provide genericfirst principle handles on QCD showered events, in connection to what (Staircase/Poisson) scaling behaviourprovides for the rates. A. General properties of splitting scales
In order to discuss splitting scales as derived from the Gen- k t generating functional, we first note that emissionare separately ordered in energy and angle in this framework, and not explicitly in k t . We can enforce k t orderingby hand, with a Θ function for example, bearing in mind that zE (cid:112) ξ/ξ R corresponds to the usual definition ofthe splitting scale for a k t jet algorithm [5, 33] in the small- R limit.Our notation for the splitting scales is that (cid:104) k ( i,j ) t (cid:105) is the j hardest emission in k t in a i jet splitting history, sothat we will always speak of the distribution or ordered scales for an exclusive event. Now let us discuss brieflywhat we expect the resulting distributions to convey regarding the nature of QCD showering.First of all, since we know that the Pseudo-abelian limit where all emissions are primary corresponds to aPoisson process, the average intermediate scale (cid:104) k ( i ) t (cid:105) is independent of the multiplicity. This property is trueboth at the inclusive and exclusive level, indicating that it persists for fixed-order and resummed calculations.A way to see that this is that the splitting scales correspond to the inter-arrival times of the Poisson process,which are themselves memory-less. R . . . . . .
09 0 .
08 0 .
07 0 .
06 0 .
05 0 . | log(∆( e, ξ )) | . . . . . . . . . . . n . . . . . . . . . . . R .
75 0 .
79 0 .
82 0 .
857 0 .
921 0 .
934 0 .
937 0 .
938 0 .
941 0 .
945 0 . (cid:12)(cid:12) dR dn (cid:12)(cid:12) .
018 0 .
010 0 . . . . . . . . . R . The first two rows show theexpected and observered Poisson parameter ¯ n . The last two rows contain the numerical value for R compared to dR /dn . The constant average splitting scales in the pseudo-Abelian limit is similar in origin to the idealised σ n +1 /σ n =1 / ( n +1) behaviour in the rates for a Poisson process. In QCD, once we allow for correlated emissions, the picturewill change, and we may ask the question whether there is some notion of a correlated emission dominated phaseimprinted in the splitting scales. As we will see, our computations indicate that there is some evidence forbehaviour like this. One emission
We start with the single emission splitting scale. This is the average value of Ez (cid:112) ξ (cid:48) /ξ R of the probabilityfunction ( zξ (cid:48) ) − α S ( zEξ (cid:48) ) ∆( z, ξ (cid:48) ) in the plane defined by boundaries ξ (cid:48) ∈ [ ξ R , ξ ] and z ∈ [1 /e, (cid:104) k t (cid:105) = zE (cid:112) ξ (cid:48) /ξ R inside the integral is (cid:104) k (1 , t (cid:105) = E (cid:82) ξξ R dξ (cid:48) √ ξ (cid:48) (cid:82) /e dz z P ( z ) α S ( zEξ (cid:48) ) ∆( z, ξ (cid:48) ) √ ξ R (cid:82) ξξ R dξ (cid:48) ξ (cid:48) (cid:82) /e dz P ( z ) α S ( zEξ (cid:48) ) ∆( z, ξ (cid:48) ) (20)which keeping only the most singular contributions gives the leading logarithms (cid:104) k (1 , t (cid:105) = 2 ( E − E R )( (cid:112) ξ/ξ R − E/E R ) log( (cid:112) ξ/ξ R ) + O ( α S ) (21)The expression in Eq. (20) is well approximated by Eq. (21) for all but large energy scale ratios O (10 ), orextremely small jet radii. This effect can be seen by noting that the inclusion of higher order terms in theSudakov affects the numerator and denominator of Eq. (20) in the same direction, thus leaving a diminishedresidual dependence. Furthermore, there is no leading order dependence on α S , a consequence is that runningcoupling effects are pushed to third-order. Two emissions
In order to compute the average splitting scales for two emission we distinguish between k (21) t and k (22) t suchthat we have k (21) t > k (22) t . Furthermore, at this multiplicity there are two splitting histories which we denotecorrelated and uncorrelateduncorrelated : correlated :We write the two primary emission rate in a k t ordered way such that the contribution to lowest order is P (uncorr)2 = 12 α S C F π (cid:32)(cid:90) ξξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz P ( z ) (cid:90) ξξ R dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) (cid:90) E R /E dz (cid:48) P ( z (cid:48) ) Θ( z (cid:48) (cid:112) ξ (cid:48)(cid:48) < z (cid:112) ξ (cid:48) )+ (cid:90) ξξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz P ( z ) (cid:90) ξξ (cid:48) dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) (cid:90) E R /E dz (cid:48) P ( z (cid:48) ) Θ( z (cid:48) (cid:112) ξ (cid:48)(cid:48) ≥ z (cid:112) ξ (cid:48) ) (cid:33) (22)0 E R √ ξ R E (cid:104) k (1) t (cid:105)(cid:104) k (22) t (cid:105) u (cid:104) k (21) t (cid:105) u (cid:104) k (21) t (cid:105) c (cid:104) k (22) t (cid:105) c ∆ k ∆ k · · · Figure 4: Schematic distribution for the average splitting scales (cid:104) k t (cid:105) for 1 and 2 gluon emissions, in the latter case splitin terms of the correlated and uncorrelated component. The scale E represents the initial energy of the emitting parton. which gives the well known result (1 / P for the rate. The contributions in Eq. (22) arise when the secondemission in the integral is smaller or larger in k t respectively. For correlated emissions, due to ordering only thefirst term is present.The Θ function provides a complicated phase space constraint, preventing a simple analytic evaluation. Inthe inclusive k t (Durham) algorithm, the k t regions would factorise, and we would expect a much simpler finalresult. However, in this work we have stayed as close as possible to the more typically used jet variable E R and ξ R . Despite this complication, we can extract the average splitting scale of the harder emission (cid:104) k (21) t (cid:105) uncorr. = EP (uncorr)2 α S C F π (cid:32)(cid:90) ξξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz P ( z ) z (cid:112) ξ (cid:48) (cid:90) ξξ R dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) (cid:90) E R /E dz (cid:48) P ( z (cid:48) ) Θ( z (cid:48) (cid:112) ξ (cid:48)(cid:48) < z (cid:112) ξ (cid:48) )+ (cid:90) ξξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz P ( z ) (cid:90) ξξ (cid:48) dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) (cid:90) E R /E dz (cid:48) P ( z (cid:48) ) z (cid:48) (cid:112) ξ (cid:48)(cid:48) Θ( z (cid:48) (cid:112) ξ (cid:48)(cid:48) > z (cid:112) ξ (cid:48) ) (cid:33) , (23)and similarly for the softer (cid:104) k (22) t (cid:105) uncorr. = EP (uncorr)2 α S C F π (cid:32)(cid:90) ξξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz P ( z ) (cid:90) ξξ R dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) (cid:90) E R /E dz (cid:48) P ( z (cid:48) ) z (cid:48) (cid:112) ξ (cid:48)(cid:48) Θ( z (cid:48) (cid:112) ξ (cid:48)(cid:48) < z (cid:112) ξ (cid:48) )+ (cid:90) ξξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz P ( z ) z (cid:112) ξ (cid:48) (cid:90) ξξ (cid:48) dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) (cid:90) E R /E dz (cid:48) P ( z (cid:48) ) Θ( z (cid:48) (cid:112) ξ (cid:48)(cid:48) > z (cid:112) ξ (cid:48) ) (cid:33) , (24)We can evaluate these analytically by taking only the energy ordering enforced by the Θ functions, which differsfrom the full result by sub-leading terms, and demonstrates the same essential behaviour. In this case we obtain (cid:104) k (21) t (cid:105) uncorr. = 4log ( E/E R ) log( ξ/ξ R ) (cid:18) E R − E + E log (cid:20) EE R (cid:21)(cid:19) ( (cid:112) ξ/ξ R − , (cid:104) k (22) t (cid:105) uncorr. = (cid:104) k (21) t (cid:105) uncorr. E − E R + E R log( E R /E ) E R − E + E log( E/E R ) . This also gives the symmetric fluctuation about the uncorrelated average in the individual components as ∆ k = (cid:104) k (1) t (cid:105) − (cid:104) k (22) t (cid:105) (see Fig. 4).Turning to the correlated emissions, due to the energy and angular ordering in the correlated emission, thiscontribution is easily evaluated and gives (cid:104) k (21) t (cid:105) corr. = EP (cid:48) α S C F C A π (cid:90) ξξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz P ( z ) z (cid:112) ξ (cid:48) (cid:90) ξ (cid:48) ξ R dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) (cid:90) zE R /E dz (cid:48) P ( z (cid:48) )= 4log ( E/E R ) log ( ξ/ξ R ) (cid:18) E R − E + E R log (cid:20) EE R (cid:21)(cid:19) (cid:18) − (cid:112) ξ/ξ R + (cid:112) ξ/ξ R log (cid:20) ξξ R (cid:21)(cid:19) , (25)and (cid:104) k (22) t (cid:105) corr. = E EP (cid:48) α S C A C F π (cid:90) ξξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz P ( z ) (cid:90) ξ (cid:48) ξ R dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) (cid:90) zE R /E dz (cid:48) P ( z (cid:48) ) z (cid:48) (cid:112) ξ (cid:48)(cid:48) = 4log ( E/E R ) log ( ξ/ξ R ) (cid:18) E − E R + E R log (cid:20) E R E (cid:21)(cid:19) (cid:18) (cid:112) ξ/ξ R − − (cid:20) ξξ R (cid:21)(cid:19) . (26)1 (cid:88) k (cid:72) (cid:76) (cid:92) uncorr (cid:88) k (cid:72) (cid:76) (cid:92) uncorr Av. uncorr (cid:88) k (cid:72) (cid:76) (cid:92) corr (cid:88) k (cid:72) (cid:76) (cid:92) corr Av. corr k t E √ ξ R R uncorrelated2 emission correlated3 emission correlated4 emission correlated (cid:104) k ( i ) t (cid:105)(cid:104) k (1) t (cid:105) R Figure 5: (Left) Distribution for the scaled average splitting scale (cid:104) k t (cid:105) /E for 2 gluon emission as a function of R , dividedamong the correlated and uncorrelated components. (Right) Ratio of the average splitting scale per jet for 1,2,3,4 emissionsfor the all correlated contribution. Three emissions
For the 3 emission component we proceed as before though we have 4 splitting histories, which differ by theirsequence of primary (P) and secondary (S) emissions. They can be written
P P P , P P S , P SP and
P SS wherePPP : PPS : PSP : PSS :First we have the normal Poisson relation that (cid:104) k (32) t (cid:105) P P P = (cid:104) k (1) t (cid:105) and (cid:104) k (31) t (cid:105) P P P − (cid:104) k (32) t (cid:105) P P P = (cid:104) k (32) t (cid:105) P P P −(cid:104) k (33) t (cid:105) P P P in the all correlated part. We give here only the analytic form of the
P SS (or fully correlated)contribution for the hardest emission (cid:104) k (31) t (cid:105) PSS = 288log ( E/E R ) log ( ξ/ξ R ) E R (cid:18) E log (cid:18) EE R (cid:19) ( E R − E ) log (cid:18) EE R (cid:19) + 4 E − E R ) − E + 2 E R (cid:19)(cid:18)(cid:112) ξ/ξ R (cid:18)
18 log (cid:18) ξξ R (cid:19) −
12 log (cid:18) ξξ R (cid:19) + 1 (cid:19) − (cid:19) . (27)As with the 2 correlated emissions the average of the splitting scale for 3 correlated emissions ( (cid:104) k (31) t (cid:105) + (cid:104) k (32) t (cid:105) + (cid:104) k (33) t (cid:105) P SS ) / (cid:104) k (1) t (cid:105) . Results for all multiplicity
In principle we can repeat the procedure for an arbitrary splitting history, and thus for all multiplicities.However, the current method is extremely tedious. Instead we will focus only on the all correlated contributionsince it has the largest deviation from a Poissonian splitting distribution for a fixed multiplicity. In other words,all average splitting scales for an arbitrary history fall somewhere between the all correlated and the all primaryresults. The leading contribution to the all correlated average at fixed coupling is obtained from the limit (cid:104) k ( ∞ ) t (cid:105) = lim j →∞ (cid:16) E (cid:81) jk =1 (cid:82) ξ k +1 ξ R dξ k ξ k (cid:82) z k +1 E R /E dz k z k (cid:17) (cid:80) jk =1 z k (cid:112) ξ k /ξ R j (cid:16)(cid:81) jk =1 (cid:82) ξ k +1 ξ R dξ k ξ k (cid:82) z k +1 E R /E dz k z k (cid:17) (28)which for the general case we are not able to take analytically. However a trick at our disposal is the small R limit, where the small R behaviour of the correlated plots in Fig. 5 suggests that this converges to a finitenon-zero value. We find the following series representation for Eq. (28) divided by the single emission average2in that case, (cid:104) k ( ∞ ) t (cid:105)(cid:104) k (1) t (cid:105) = lim j →∞ ( j + 1)!(1 − e ) log j ( e ) (cid:32) − e + j (cid:88) k =1 log k ( e ) k ! (cid:33) + O (1 / log(1 /R )) . (29)Substituting the asymptotic Stirling approximation for the the sum in brackets, we find the surprisingly simpleexpression (cid:104) k ( ∞ ) t (cid:105)(cid:104) k (1) t (cid:105) = lim j →∞ ( j + 1)!(1 − e ) log j ( e ) (cid:32) − log j +1 ( e ) j j +1 exp( − j ) √ πj (cid:33) + O (1 / log(1 /R ))= log( e ) e − R behaviour of the correlated emission scale (cid:104) k ( ∞ ) t (cid:105) = E/ log( √ /R ), which canbe compared to the pure Poisson single emission result at small- R , given from Eq. (21) by (cid:104) k (1) t (cid:105) = E (1 − e ) / (log( e ) log( √ /R )) Note that these expression very nicely enforce the intuitive fact that (cid:104) k (1) t (cid:105) < (cid:104) k ( ∞ ) t (cid:105) forall values of E R and R , even though the latter does not depend on E R explicitly. The correlated emissioncomponent, via the precise cancellation in Eq. (29) in the R (cid:28) E R . B. Discussion of results
We summarise the findings of our calculations for the average k t splitting scales. For arbitrary multiplicity,the all uncorrelated contribution produces a Poisson process where the average splitting scale is a constantfor all n . The largest deviation from a Poisson process is the all correlated emission component, which putsthis observable in a similar class as the average jet multiplicity [34], in the sense of being maximally sensitiveto correlated emissions. We were able to show that at leading order in the small- R limit, the all-correlatedcontribution converges to a constant. Of course physically it must converge to something since it is boundedfrom above and below, but what is surprising is the simple analytic result in this case. We speculate that forthe generic leading logarithmic QCD behaviour of the average jet multiplicity it converges to a constantlim n →∞ (cid:104) ¯ k ( n ) t (cid:105)(cid:104) ¯ k (1) t (cid:105) = 1 + c, (31)with c >
0. Whether or not this behaviour is realised in realistic QCD environments, for example in the internaldynamics of large jets, is a question which can be answered with a dedicated phenomenological study.
IV. CONCLUSIONS
In this paper we have studied jets at high multiplicity. Using the Gen- k t generating functional, we found thatscaling patterns in the rates emerged in particular limits in energy, angle and multiplicity. We speculated thatfinite area effects are responsible for depressing the geometric scaling in the jet ratios and attempted to quantifythe area distribution as accurately as possible. Using these considerations we managed to describe the areadistribution coming from a parton shower simulation. The area depletion effect on the rates on the other hand,was found to be significant but too small to explain the tilt in the n -jets distribution at finite R , indicating thatadditional terms are necessary to fully, analytically describe the parton shower generated n jet distribution. Inthe second part of this paper we looked into a new observable from the perspective of scaling and found thatit could also be shown, formally at least, to produce a type of universal behaviour at high multiplicity, whichemerges due to the dominance of correlated emissions in this limit.We did not undertake a detailed phenomenological study in this work. However, one can imagine that sub-jetmultiplicities with an energy cut sufficiently small might also show an extended regime of generalised types ofscaling. Furthermore, the distribution of jet areas can be extended to regimes where this quantity is useful,for example assessing pile-up contributions and non-perturbative effects. Finally, we speculate that additionaldistributions and observables may show surprising emergent properties at high multiplicity, which could lead toexperimental and theoretical progress.3 Acknowledgments
We thank Steffen Schumann for helpful comments on the draft. PS acknowledges support from the europeanUnion as part of the FP7 Marie Curie Initial Training Network MCnet ITN (PITN-GA-2012-315877) and theIMPRS for Precision Tests of Fundamental Symmetries. EG acknowledges support by the Bundesministeriumf¨ur Bildung und Forschung under contract 05H2012.
Appendix A: Exponentiated form of the evolution equation
We start from Eq. (4.7) in [10], which isΦ q ( E, ξ ) = u + ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz α s ( k T )2 π P q → qg Φ q ( E, ξ (cid:48) ) (Φ g ( zE, ξ (cid:48) ) − g ( E, ξ ) = u + ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) E R /E dz α s ( k T )2 π [ P g → gg Φ g ( E, ξ (cid:48) ) (Φ g ( zE, ξ (cid:48) ) − P g → q ¯ q (cid:0) Φ q ( E, ξ (cid:48) ) − Φ g ( E, ξ (cid:48) ) (cid:1)(cid:3) . (A1)Using e = E/E R we rewrite it in the formΦ i ( e, ξ ) = u + ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) /e dz α s ( ze, ξ (cid:48) )2 π (cid:88) j,k P i → jk ( z )Φ i ( e, ξ (cid:48) ) (cid:18) Φ j ( e, ξ (cid:48) )Φ k ( E ( z ) , ξ (cid:48) )Φ i ( e, ξ (cid:48) ) − (cid:19) , (A2)where the running coupling is expressed as follows [10] α s ( k T ) = πb log z e E R ξ Λ . (A3) i is either q or g and we sum over all allowed splittings. Now we take the derivative with respect to ξ and find d Φ i ( e, ξ ) dξ = 1 ξ Φ i ( e, ξ ) (cid:90) /e dz α s ( ze, ξ )2 π (cid:88) j,k P i → jk ( z ) (cid:18) Φ j ( e, ξ )Φ k ( E ( z ) , ξ )Φ i ( e, ξ ) − (cid:19) . (A4)Taking into account that Φ i ( e,
1) = u the general closed solution isΦ i ( e, ξ ) = u exp ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) /e dz α s ( ze, ξ (cid:48) )2 π (cid:88) j,k P i → jk ( z ) (cid:18) Φ j ( e, ξ )Φ k ( E ( z ) , ξ )Φ i ( e, ξ ) − (cid:19) . (A5) Appendix B: Closed solution in the staircase limit
We start from Eq. (8)Φ g ( e, ξ ) = u exp ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) /e dz α s ( z, ξ (cid:48) )2 π P g → gg ( z ) (cid:34) Φ g ( e, ξ (cid:48) ) + ∞ (cid:88) n =1 ( e ( z − n n ! d n Φ g ( e, ξ (cid:48) ) de n − (cid:35) . (B1)Note that due its exponantial form all derivatives of Φ g can be written in the following form d n Φ g ( e, ξ (cid:48) ) de n = Φ g ( e, ξ (cid:48) ) × DP [ n ]( e, ξ (cid:48) ; Φ g ) , (B2)4where DP [ n ] is a polynomal of inner derivatives of Φ g . We plug this in eq. (B1) and getΦ g ( e, ξ ) = u exp ξ (cid:90) ξ R dξ (cid:48) ξ (cid:48) (cid:90) /e dz α s ( z, ξ (cid:48) )2 π × P g → gg ( z ) Φ g ( e, ξ (cid:48) ) + Φ g ( e, ξ (cid:48) ) ∞ (cid:88) n =1 ( e ( z − n n ! DP [ n ]( e, ξ (cid:48) ; Φ g ) (cid:124) (cid:123)(cid:122) (cid:125) T ( z,e,ξ (cid:48) ) − . (B3)Up to this point we have not gained any new insight. We can now introduce more new symbolds to rewriteeq. (B3) and bring it in the formΦ g ( e, ξ ) = u exp ξ (cid:90) ξ R dξ (cid:48) (Φ g ( e, ξ (cid:48) ) −
1) 1 ξ (cid:48) (cid:90) /e dz α s ( z, ξ (cid:48) )2 π P g → gg ( z ) (cid:124) (cid:123)(cid:122) (cid:125) γ g ( e,ξ (cid:48) ) + ξ (cid:90) ξ R dξ (cid:48) Φ g ( e, ξ (cid:48) ) 1 ξ (cid:48) (cid:90) /e dz α s ( z, ξ (cid:48) )2 π P g → gg ( z ) T ( z, e, ξ (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) r ( e,ξ (cid:48) ) = u exp ξ (cid:90) ξ R dξ (cid:48) (Φ g ( e, ξ (cid:48) ) − γ g ( e, ξ (cid:48) ) + ξ (cid:90) ξ R dξ (cid:48) Φ g ( e, ξ (cid:48) ) r ( e, ξ (cid:48) ) . (B4)Taking the derivative this defines a differential equation of the form d Φ g ( e, ξ ) dξ = Φ g ( e, ξ ) × [ γ g ( e, ξ ) (Φ g ( e, ξ ) −
1) + r ( e, ξ )Φ g ( e, ξ )]Φ g ( e, ξ r ) = u. (B5)We note that for the case r ( e, ξ ) → r ( e, ξ ). The solution isΦ g ( e, ξ ) = 11 + (1 − u ) u ∆ g ( e,ξ ) − ξ (cid:90) ξ R dξ (cid:48) ∆ g ( e, ξ (cid:48) )∆ g ( e, ξ ) r ( e, ξ (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) R ( e,ξ ) (B6)where ∆ g ( e, ξ ) = exp (cid:34) − ξ (cid:82) ξ R dξ (cid:48) γ ( e, ξ (cid:48) ) (cid:35) is the Sudakov form factor. This is a neat result. We find that we canwrite the generating functional in general in a staircase like form together with an yet unspecified staircasescaling breaking term. We check the solution explicitly by differentiating eq. (B6) with respect to ξ . We notethat we have d ∆ g ( e, ξ ) dξ = − γ ( e, ξ )∆ g ( e, ξ ) d R ( e, ξ ) dξ = r ( e, ξ ) + γ ( e, ξ ) R ( e, ξ ) . (B7)5Therefore, we get d Φ g ( e, ξ ) dξ = ( − (cid:18) − uu ∆ g ( e, ξ ) − R ( e, ξ ) (cid:19) − × (cid:20) − uu ∆ g ( e, ξ ) γ ( e, ξ )∆ g ( e, ξ ) − r ( e, ξ ) − γ ( e, ξ ) R ( e, ξ ) (cid:21) = Φ g ( e, ξ ) × [ γ g ( e, ξ ) (Φ g ( e, ξ ) −
1) + r ( e, ξ )Φ g ( e, ξ )] . (B8)To evolve further we need to employ some assumptions. This means we have to drop the explicit Φ g dependencein T . For example we could expand around e ≈ g as an explicit series in u , dropping all higherterms in e . To find the leading u dependence we plug in all previous definitions and find that R ( e, ξ ) = ξ (cid:90) ξ R dξ (cid:48) ∆ g ( e, ξ (cid:48) )∆ g ( e, ξ ) r ( e, ξ (cid:48) )= ξ (cid:90) ξ R dξ (cid:48) ∆ g ( e, ξ (cid:48) )∆ g ( e, ξ ) 1 ξ (cid:48) (cid:90) /e dz α s ( z, ξ (cid:48) )2 π P g → gg ( z ) T ( z, e, ξ (cid:48) )= ξ (cid:90) ξ R dξ (cid:48) ∆ g ( e, ξ (cid:48) )∆ g ( e, ξ ) 1 ξ (cid:48) (cid:90) /e dz α s ( z, ξ (cid:48) )2 π P g → gg ( z ) ∞ (cid:88) n =1 ( e ( z − n n ! DP [ n ]( e, ξ (cid:48) ; Φ g ) . (B9)The task at hand is to find a Φ g independent approximation for DP [ n ]. An obvious step is to truncate theTayler expansion at n = 0. We are then left with DP [1]( e, ξ (cid:48) ; Φ g ) = d Φ g ( e, ξ (cid:48) )Φ g ( e, ξ (cid:48) ) de . (B10)We need to find the significant part in the limit e → u dependence. We have DP [1]( e, ξ (cid:48) ; Φ g ) = ξ (cid:48) (cid:90) ξ R dξ (cid:48)(cid:48) ξ (cid:48)(cid:48) α s (1 /e, ξ (cid:48)(cid:48) ) P g → gg (1 /e )2 π e Φ g (1 , ξ (cid:48)(cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) = u − + (cid:90) /e dz α s ( z, ξ (cid:48)(cid:48) ) P g → gg ( z ) z π (cid:20) d Φ( e, ξ (cid:48)(cid:48) ) de (cid:12)(cid:12)(cid:12)(cid:12) e = ze (cid:21) . (B11)In principle we get a nested series of Φ g differentiations. To argue that first term is the most important we notethat formally in the e → (cid:90) /e dz α s ( z, ξ (cid:48)(cid:48) ) P g → gg ( z ) z π (cid:20) d Φ( e, ξ (cid:48)(cid:48) ) de (cid:12)(cid:12)(cid:12)(cid:12) e = ze (cid:21) ≈ (1 − /e ) α s (1 /e, ξ (cid:48)(cid:48) ) P g → gg (1 /e )2 π e (cid:20) d Φ( e, ξ (cid:48)(cid:48) ) de (cid:12)(cid:12)(cid:12)(cid:12) e ≈ (cid:21) . (B12)We note that the last term of eq. (B12) is exactly zero in the formal limit e →
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