We reconsider the calculation of a non-global QCD observable and find the
possible breakdown of QCD coherence. This breakdown arises as a result of wide
angle soft gluon emission developing a sensitivity to emission at small angles
and it leads to the appearance of super-leading logarithms. We use the `gaps
between jets' cross-section as a concrete example and illustrate that the new
logarithms are intimately connected with the presence of Coulomb gluon
contributions. Numerical estimates of their potential phenomenological
significance are presented.Comment: 7 pages, 3 figures; proceedings of Workshop on Diffraction in
  High-Energy Physics - DIFFRACTION 2006, Sep. 5-10 2006, Adamantas, Milos
  island, Greec

Forshaw, J. R.

Kyrieleis, A.

Seymour, M. H.

English

arXiv

We reconsider the calculation of a non-global QCD observable and find the possible breakdown of QCD coherence. This breakdown arises as a result of wide angle soft gluon emission developing a sensitivity to emission at small angles and it leads to the appearance of super-leading logarithms. We use the `gaps between jets' cross-section as a concrete example and illustrate that the new logarithms are intimately connected with the presence of Coulomb gluon contributions. Numerical estimates of their potential phenomenological significance are presented

Kyrieleis, A

Forshaw, J R

Seymour, M H

CERN Document Server

PoS(DIFF2006)031Breakdown of QCD coherence ?A. Kyrieleis∗University of Manchester, U.K.E-mail: kyrieleis@hep.man.ac.ukJ.R. ForshawUniversity of Manchester, U.K.E-mail: forshaw@mail.cern.chM.H. SeymourCERN and University of Manchester, U.K.E-mail: mike.seymour@cern.chWe reconsider the calculation of a non-global QCD observable and find the possible breakdownof QCD coherence. This breakdown arises as a result of wide angle soft gluon emission devel-oping a sensitivity to emission at small angles and it leads to the appearance of super-leadinglogarithms. We use the ‘gaps between jets’ cross-section as a concrete example and illustrate thatthe new logarithms are intimately connected with the presence of Coulomb gluon contributions.Numerical estimates of their potential phenomenological significance are presented.International Workshop on Diffraction in High-Energy Physics -DIFFRACTION 2006 -September 5-10 2006Adamantas, Milos island, Greece∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(DIFF2006)031Breakdown of coherence? A. Kyrieleis1. IntroductionThe resummation of large logarithms associated with wide angle soft gluon emissions hasbeen investigated for the last 20 years. For certain observables the contributions from non-globallogarithms [1] have to be taken into account. One of the simplest of these non-global observablesis the ‘gaps-between-jets’ cross-section. This is the cross-section for producing a pair of hightransverse momentum jets (Q) with a restriction on the transverse momentum of any additional jetsradiated in between the two jets, i.e. kT < Q0 for emissions in the gap. This observable has beenstudied [2, 3] and has been measured at HERA and the Tevatron [4].In the original calculations [2] of the gaps-between-jets cross section, all terms proportional toαns lnn(Q2/Q20) that can be obtained by dressing the primary 2 → 2 scattering in all possible wayswith soft virtual gluons were summed. The restriction to soft gluons implies the use of the eikonalapproximation. Let us focus on quark-quark scattering from now on. The corresponding resummedcross-section can be writtenσ = M†SV M with M = exp−2αspiQ∫Q0dkTkTΓM0. (1.1)Here, M is the all-orders qq → qq amplitude (a 2-component vector in colour space), M0 is thehard scattering amplitude and SV represents the cut. The anomalous dimension matrix Γ [5] in-corporates the effect of dressing a qq → qq amplitude with a virtual gluon in all possible ways.It receives contributions from two distinct regions of the loop-integral: the first corresponds to anon-shell gluon (to which one can assign a rapidity) and is identical, but with opposite sign, to thecontribution from a real gluon. The second contribution, sometimes referred to as the ‘Coulombgluon contribution’ [6] is purely imaginary (ipi terms) and stems from the region where the emittingparton is on-shell. Eq. (1.1) therefore corresponds to the independent emission of soft gluons, i.e.the iterative dressing of the 2 → 2 process with a softer gluon: due to perfect real/virtual cancel-lation outside the gap (the first line of Fig. 1 shows two contributions) one only has to considervirtual gluons in the gap and the Coulomb terms.Figure 1: Illustrating the cancellation (and miscancellation) of soft gluon corrections.However, there is another source of leading logarithms. Let us consider the two diagrams inthe second line of Fig. 1. A real gluon (which is outside the gap by the definition of our observable)2PoS(DIFF2006)031Breakdown of coherence? A. Kyrieleisemits a softer real or virtual gluon. The real-virtual cancellation is guaranteed only for the softestgluon. Since real gluons above Q0 are forbidden in the gap, the two diagrams do not completelycancel; the left diagram with the virtual gluon being in the gap and its kT being larger than Q0survives. The non-global nature of our observable has prevented the soft gluon cancellation whichis necessary in order that Eq. (1.1) should be the complete result.It is therefore necessary to include the emission of any number of soft gluons outside the gapregion (real and virtual) dressed with any number of virtual gluons within the gap region. Clearly itis a formidable challenge to sum all leading logarithms, mainly because of the complicated colourstructure. Progress has been made, working in the large N approximation [3]. Here, we keep theexact colour structure but instead we only compute the cross-section for one gluon outside the gapregion. This can be viewed as the first term in an expansion in the number of out-of-gap-gluons.2. Super-leading logarithmsIn order to extract the leading logarithms we consider soft gluons strongly ordered in transversemomentum. The cross-section for one gluon outside and any number of gluons inside the gap issplit into two parts corresponding to a virtual or real out-of-gap gluon:σ1 =−α¯∫ QQ0dkTkT∫outdy dφ2pi(ΩV +ΩR) , α¯ ≡2αspi(2.1)ΩR = M†0 exp−α¯Q∫kTdk′Tk′TΓ†D†µ exp−α¯kT∫Q0dk′Tk′TΛ†SRexp−α¯kT∫Q0dk′Tk′TΛDµ exp−α¯Q∫kTdk′Tk′TΓM0 , (2.2)ΩV = M†0 exp−α¯Q∫Q0dk′Tk′TΓ†SV exp−α¯kT∫Q0dk′Tk′TΓ γ exp−α¯Q∫kTdk′Tk′TΓM0 + c.c. (2.3)Dµ and γ are the matrices that represent the emission of a real and a virtual gluon (kT ,y,φ ) outsidethe gap, respectively. The major new ingredient is the matrix Λ [7] which incorporates the dressingof the qq → qqg process with a virtual gluon. The emission of the out-of-gap gluon is sandwichedbetween two exponentials: this accounts for all possible positions of the out-of-gap gluon within achain of any number of kT -ordered gluons within the gap.The phase space of the out-of-gap gluon in Eq. (2.1) includes the configurations where it iscollinear to either of the external quarks. One might suppose that the corresponding divergencescancel among ΩR and ΩV . This is true in case of the final state collinear limit. However, in the limitof the out-of-gap gluon becoming collinear to one of the initial state quarks, which corresponds to|y| → ∞,kT > Q0, there is no cancellation:[ΩV +ΩR]|y|→∞ 6= 0. (2.4)3PoS(DIFF2006)031Breakdown of coherence? A. KyrieleisIn particular, (ΩV + ΩR) becomes independent of y in that limit. This has severe consequences.As the out-of-gap region stretches to infinity in rapidity, the integral Eq. (2.1) is divergent as itstands. This divergence however indicates that one needs to go beyond the soft approximationwhen considering the out-of-gap gluon. Strictly speaking we ought to work in the collinear (butnot soft) approximation which means that the integral over rapidity ought to be replaced by∫d2kT∫outdy dσdyd2kT∣∣∣∣soft→∫d2kTymax∫dy dσdyd2kT∣∣∣∣soft+∞∫ymaxdy dσdyd2kT∣∣∣∣collinear . (2.5)In this equation ymax is a matching point between the regions in which the soft and collinear approx-imations are used. If ymax is in the region in which both approximations are valid the dependenceon it should cancel in the sum of the two terms. Now we know that∞∫ymaxdy dσdyd2kT∣∣∣∣collinear=∞∫ymaxdy(dσRdyd2kT∣∣∣∣collinear+dσVdyd2kT∣∣∣∣collinear)(2.6)where the contribution due to real gluon emission can be written as∞∫ymaxdy dσRdyd2kT∣∣∣∣collinear=1−δ∫0dz12(1+ z21− z)(q(x/z,µ2)q(x,µ2) −1)AR +1−δ∫0dz121+ z21− zAR (2.7)and the contribution due to virtual gluon emission is∞∫ymaxdy dσVdyd2kT∣∣∣∣collinear=1−δ∫0dz12(1+ z21− z)AV. (2.8)In Eq. (2.7), q(x,µ2) is the parton distribution function for a quark in a hadron at scale µ2 andmomentum fraction x. The factors AR and AV contain the z independent factors which describethe soft gluon evolution. Since we require y > ymax1 the upper limit on the z integral is fixed:δ ≈ kT /Q · exp(ymax − ∆y/2). We have already established that AR + AV 6= 0 due to Coulombgluon contributions to the evolution. If it were the case that AR +AV = 0 then the virtual emissioncontribution would cancel identically with the corresponding term in the real emission contributionleaving behind a term regularised by the ‘plus prescription’ (since we can safely take δ → 0 in thefirst term of Eq. (2.7)). This term could then be absorbed into the evolution of the incoming quarkparton distribution function by choosing the factorisation scale to equal the jet scale Q.The miscancellation therefore induces an additional contribution of the form1−δ∫0dz12(1+ z21− z)(AR +AV) = ln(1δ)(AR +AV)+ subleading (2.9)≈(−ymax +∆y2+ ln( QkT))(AR +AV). (2.10)1The approximation arises since we assume for simplicity that ∆y is large and δ is small. This approximation doesnot affect the leading behaviour and can easily be made exact if necessary.4PoS(DIFF2006)031Breakdown of coherence? A. KyrieleisProvided we stay within the soft-collinear region in which both the soft and collinear approxima-tions are valid, the ymax dependence will cancel with that coming from the soft contribution inEq. (2.5) leaving only the logarithm. The leading effect of treating properly the collinear regionis therefore simply to introduce an effective upper limit ∆y/2 + ln(Q/kT ) to the integration overrapidity in Eq. (2.5).2αspi∫ QQ0dkTkTln(Q/kT )+∆y/2∫Y/2dy dφ2pi=2αspi12ln2(Q/Q0)+ subleading. (2.11)This is the super-leading logarithm: the failure of the ‘plus prescription’ has resulted in the gener-ation of an extra collinear logarithm. The implications for the gaps-between-jets cross-section areclear: collinear logarithms can be summed into the parton density functions only up to scale Q0and the logarithms in Q/Q0 from further collinear evolution must be handled separately.The miscancellation Eq. (2.4) and hence the super-leading logarithm is intimately connectedwith the Coulomb phase terms. If one artificially switches off the ipi terms in the evolution matri-ces, then there is full cancellation in Eq. (2.4). Moreover, the super-leading logarithm makes itsappearance at the lowest possible order in αs, i.e. at O(α4s ) relative to the Born cross-section. Thisis due to the fact that at lower orders any ipi term is cancelled by a corresponding term from thecomplex conjugate contribution. The first ipi terms and the first super-leading logarithm appear incase of four soft gluons:σ1 ∼ σBorn(2αspi)4ln5( QQ0)pi2Y. (2.12)At higher orders in αs more gluons can be outside the gap. However, to resum the double loga-rithms to all orders a deeper understanding of the colour evolution of multi-parton systems seemsnecessary.Indeed we appear to have uncovered a breakdown of QCD coherence: radiation at large anglesdoes appear to be sensitive to radiation at low angles. However this striking conclusion was arrivedat under the assumption that it is correct to order successive emissions in transverse momentum.Coherence indicates that one does not need to take too much care over the ordering variable, e.g. kT ,E and k2T /E are all equally good ordering variables but the super-leading logarithms arise counterto the expectations of coherence and in particular as a result of radiation which is both soft andcollinear. It is therefore required to prove the validity of kT ordering before we can claim withoutdoubt the emergence of super-leading logarithms or confirm their size.3. Numerical resultsWe numerically compute the out-of-gap cross section which is the sum of Eq. (2.2) and Eq.(2.3) and which which we generically denote σ1. The purely superleading logarithmic part of σ1is obtained by considering the initial state collinear limit and performing the integral over rapidityover an interval of size ln(Q/kT ). The result is multiplied by 2 to account for the possibility thatthe out-of-gap gluon can be on either side of the gap. We refer to the cross section thus computedas ‘SLL’ in the figures. For comparison, we also compute the sum of Eq. (2.2) and Eq. (2.3)without making the collinear approximation. In this case the integral over y is over the region5PoS(DIFF2006)031Breakdown of coherence? A. KyrieleisY/2 < |y|< ∆y/2+ ln(Q/kT ) where ∆y =Y +2. This cross-sections is labelled ‘all’ and necessarilyincludes a partial summation of the single logarithmic terms as well as the super-leading terms. Thestrong coupling is fixed at αs = 0.15. Fig. 2 shows σ1 as a function of L = ln(Q2/Q20), normalizedto the fully resummed cross-section σ0 corresponding to zero gluons outside of the gap region, i.e.as determined by Eq. (1.1). The super-leading series is generally small relative to the ‘all’ resultfor L . 4, which indicates that the single logarithms are phenomenologically much more importantthan the formally super-leading logs at these values of L. Of course one should remember that ourcalculations are for the emission of one gluon outside the gap region and the full super-leadingseries requires the computation of any number of such gluons.From a more theoretical perspective it is interesting to take a look at the cross-sections out tolarger values of Y , see Fig. 3. Note that this time we have normalized the cross-section by thesquare of the in-gap cross-section. The cross-section saturates at large enough Y , i.e. σ1 ∼ −σ 20 .In [9] we calculated the conventional gap-between-jets cross-section in the high energy limit andshowed that it is equivalent to the BFKL result in the region in which both are valid. Here, wefind that in the high-energy (large Y ) limit the cross section for one emission outside the gap isproportional to the square of the conventional gap cross-section, offering a tantalizing clue to thestructure of higher orders. A deeper understanding of this connection would almost certainly opennew avenues to understanding non-global observables.-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0  1  2  3  4  5  6  7σ1/σ0LY=3allSLLFigure 2: L dependence of the out-of-gap cross-section (normalized to the in-gap cross-section) at Y=34. OutlookWe appear to have found the breakdown of the intuitive picture of QCD coherence: superlead-ing logarithms appear in the gaps-between-jets observable as the consequence of the sensitivityof soft wide angle gluon emission to collinear emission. The full confirmation of this findingthough requires the proof of the validity of kT -ordering. The new super-leading contributions arenot restricted to the gaps-between-jets observable. We expect them to arise generally in non-globalobservables and potentially as additional leading logarithms also in global observables where non-global contributions are subleading. The new contributions will therefore possibly have an impact6PoS(DIFF2006)031Breakdown of coherence? A. Kyrieleis-25-20-15-10-5 0 0  20  40  60  80  100  120  140σ1/σ02YSLL, L=3SLL, L=5SLL, L=7Figure 3: Y dependence of the out-of-gap cross-section normalized to the square of the in-gap cross-sectionat three different values of L and plotted out to very large Y .on a wide spectrum of processes and observables, such as eventshape variables, kT -distributionsor particle production near threshold. The connection between the super-leading logarithms andhigh-energy QCD appears to offer intriguing clues for their resummation and the understanding ofnon-global observables.References[1] M. Dasgupta and G. P. Salam, Phys. Lett. B 512, 323 (2001), JHEP 0203, 017 (2002)[2] G. Oderda, G. Sterman, Phys. Rev. Lett. 81, 3591 (1998); G. Oderda, Phys. Rev. D 61, 014004 (2000);C. F. Berger, T. Kucs and G. Sterman, Phys. Rev. D 65, 094031 (2002)[3] R. B. Appleby and M. H. Seymour, JHEP 0212, 063 (2002); R. B. Appleby and M. H. Seymour,JHEP 0309, 056 (2003)[4] M. Derrick et al [ZEUS Collaboration], Phys. 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Breakdown of QCD coherence?

Forshaw, Jeffrey R

The University of Manchester - Institutional Repository

Breakdown of QCD coherence ?

Breakdown of QCD coherence ?

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