Martin Hentschinski, Heribert Weigert, and Andreas Schafer-Extension of the Color Glass Condensate Approach to Diffractive Reactions

8/3/2019 Martin Hentschinski, Heribert Weigert, and Andreas Schafer-Extension of the Color Glass Condensate Approach to D

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arXiv:hep-ph/0509272v12

5Sep2005

Extension of the Color Glass Condensate Approach to Diffractive Reactions

Martin Hentschinski, Heribert Weigert, and Andreas Schafer1

1Institut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany

(Dated: February 2, 2008)

We present an evolution equation for the Bjorken x dependence of diffractive dissociation onhadrons and nuclei at high energies. We extend the formulation of Kovchegov and Levin by relaxingthe factorization assumption used there. The formulation is based on a technique used by Weigert todescribe interjet energy flow. The method can be naturally extended to other exclusive observables.

PACS numbers: 12.38.-t, 12.38.Cy

QCD at very high parton densities is one of the mostactive frontiers both in high-energy and nuclear physicsand one of the topics where both fields clearly profit fromclose collaboration. With the advent of the LHC in 2007this topic will further gain importance. Both the searchfor new physics in proton-proton collisions and the in-vestigation of high energy medium effects in heavy ioncollisions require a solid understanding of multiple glu-

onic interactions (at the very least for the analysis ofbackgrounds). Presently, the rapid output of precise ex-perimental data at RHIC, where the same effects shouldbe present, though less pronounced, provides the maindriving force behind new theoretical developments. Oneof the theoretically most attractive approaches is knownunder the name of color glass condensate (CGC) [1] andone of its main elements is the JIMWLK-equation de-scribing the evolution of characteristic quantities withthe squared cm energy s [2]. Our paper is based on thisapproach.

At high energies, as reached in RHIC and LHC ex-periments, most QCD observables receive strong contri-

butions from multiple soft gluon emission and multipleinteractions of hard particles with soft gluons presentin the event. A reliable and transparent method to re-sum the effects of these soft gluons on the hard lead-ing particles can be formulated by using gauge linksU(x; y) = P exp ig

xy

dz.A(z) where the trajectories

(from y to x) represent the quasiclassical paths of thehard particles while the soft gluons appear in the expo-nent. Previous work has focused on inclusive reactions.Here we demonstrate how to extend this program to ex-clusive reactions and work out the example of diffractivedissociation, where we can compare to a known limitingcase [3] that emerges if we use a factorization assump-tion as in the reduction of the JIMWLK to the Balitsky-Kovchegov (BK) [4, 5] equation.

As an example let us recall that e.g. the total crosssection of deep inelastic scattering of a virtual photon ona nuclear target can be written in terms of the Us as:

DIS(Y, Q2) =

1

0

d

d2r |2|(r2(1 )Q2)

d2b

tr[1 UxU

y

]/NcY

(1)

where |2|(r2(1 )Q2) is the probability of a photonto split into a quark-antiquark pair of size r = x y,carrying longitudinal momentum fractions and 1 ,respectively. The remaining integral over the impact pa-rameter b = (x + y)/2 yields the cross section of a qqdipole of size r. The rapidity Y = ln 1/x is taken to belarge. The gauge links Ux and Uy represent the leadinghard quark and antiquark within the virtual photon wave

function. They propagate at fixed transverse coordinatesx and y along straight lines from z = to z = :

Ux = P exp ig

dzb+(z,x, x+ = 0) . (2)

In (2) we have anticipated that the hard particles inter-act, to leading order, only with b+, the soft componentof the gauge field with rapidities below some Y0. Generi-cally A(x) = b(x) + A(x) with b+ = (x)(x) andb = b = 0; A denotes all hard fluctuations.

The averaging indicated in (1) is over the soft fields

b

+

and represents all the interactions with the targetthrough gluons softer than the original qq. As such itcontains nonperturbative information that can not di-rectly be calculated. Since this decomposition into hardand soft modes is rapidity dependent, the dipole crosssection turns rapidity dependent as well. By consideringhard corrections A one can systematically calculate theY dependence and find RG equations for the dipole crosssection. A direct approach leads to an infinite hierarchyof equations, the Balitsky hierarchy [4], which can onlybe solved after truncation. A more compact formulationin terms of a single diffusion equation can be given in afunctional language. To this end one parametrizes the

lack of knowledge about the averaging procedure by us-ing a functional ZY [U], which takes on the meaning of astatistical distribution function:

. . .Y :=

D[U] . . . ZY [U] . (3)

D[U] is a Haar measure that is normalized to 1. The s-,respectively Y-, evolution for the dipole cross section andall the other more complicated correlators in the Balitskyhierarchy is then given by the JIMWLK equation, which
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governs the evolution of the functional weight Z:

YZ[U]Y = HJIMWLK Z[U]Y . (4)

(4) describes a Fokker-Planck type diffusion problem ina functional context. The JIMWLK Hamiltonian

HJIMWLK = s

22Kxzy

Uabz

(iax

iby

+ iax

iby

)

+ iaxi

ay + i

axi

ay (5)

(integration over repeated transverse coordinates is im-plied here and below) contains a real emission partproportional to a new adjoint Wilson line Uabz thatsignals the appearance of a new gluon in the finalstate and virtual corrections that guarantee finitenessof the evolution equation. The remaining ingredients

are the kernel Kxzy =(xz)(zy)(xz)2(zy)2

and functional

derivatives iax

that respect the group valued natureof the U-fields: iax := [Uxt

a]ij/Ux,ij correspondsto the left invariant vector field on the group mani-fold, while its right invariant counterpart is given by

iax := [taUx]ij/Ux,ij = Ubax ibx.To prepare for the treatment of non-inclusive observ-

ables, we will now sketch how to recover this evolutionequation from the underlying real emission amplitudes.The treatment parallels that of jet observables in [6]. Tofacilitate this construction, let us introduce Wilson linesUY,x that are (slightly) tilted w.r.t. the lightcone. Thederivation is then based on the observation that the wholecloud of Y ordered real gluons accompanying any givennumber of hard partons that can be characterized as a

product of Wilson lines U()Y1,x1

U()Yn,xn

can be gener-ated by the application of a single operator

U[U, ] = PY2 exp

i

dY1dY2(Y1Y2)

Jixz

UabY2,zb,iY2,z

iaY1,x

(6)

with Jixz :=g42(xz)i

(xz)2the eikonal current in trans-

verse coordinate space. The fields represent the glu-onic final states. The derivatives now act as iaY,x :=[UY,xta]ij/UY,x,ij; Y-ordering is such that the hard-est gluon is rightmost. This ensures that gluons can onlyemit softer ones. For DIS, the hard seed consists of a qqpair represented by a product of Wilson lines UY,xU

Y,y

at projectile rapidities. [The external color indices are

those of the amplitude.] Diagrammatically we get

U[U, ]UY,xUY,y =

# of gluons &

allowed insertions

|

|

|

|

|

|

|

|

(7)

where the vertical dashed line denotes the final state,where each line ends in a factor . These stand for explic-itly resolved partons in an interval [Y0, Y] over which we

follow the logarithmically enhanced contributions. Theremaining gluons indicate soft interactions with the tar-get below Y0, which build up the U fields. They corre-spond to the initial condition of the evolution process. Torecover the real emission part of the dipole cross section,we need to square this amplitude and integrate over phasespace for the resolved gluons. The average over the soft,unresolved gluons is done separately in amplitude and

complex conjugate amplitude [7]: we distinguish corre-sponding eikonal factors U and U. The average over theresolved final states can be made explicit by averagingover the final state variables with a Gaussian weight.For an arbitrary functional F[] this is expressed as [6]

F[] = exp

1

2

iM

i

F[]

=0

. (8)

The -correlator Ma,iY1u

b,jY2v

= a,iY1u

b,jY2v, appropriatelynormalized, is given by

Ma,iY1ub,jY2v

= 4abij(2)u,vY1,Y2 (YY1)(Y1 Y0) . (9)

M is diagonal in coordinates and rapidities and restrictedto the resolved evolution interval [Y0, Y]. In the exponentof (8) integration and summation over all indices is un-derstood. The resolved contribution of real emissions tothe dipole cross section is thus obtained by the Gaussianaverage (8) over the functional

Greal[] = U[U, ]U[U , ]tr[1 (UU)Y,x(UU)

Y,y]

Nc(10)

where NFxy = tr[1 (UU)Y,x(UU)

Y,y]/Nc is the dipole

operator of the total cross-section. No matter how the av-erage over the non-resolved modes below Y0 is achieved,the evolution of the complete real emission part is deter-mined by the Y dependence of the resolved contributions:

YGreal[] =

e12

i

M i

2

iYM(Y)

iG[]

=0

= U[U, ]U[U , ]s2

Kxzy(UU)abY,zi

aUY,x

ibUY,y

NFxy

.

(11)

In this equation everything outside the shower operatorsonly contains U factors at the upper Y limit. This allowsto recast both of these averages in terms of averages overWilson lines at this highest rapidity: one can set

...Y = U[U, ]U[U , ]...,soft =

D[U]D[U]...ZY [U, U]

(12)

We may drop the now unnecessary Y-label on the Wilsonlines. This result, as all inclusive quantities, only dependson products UU in the hard operators appearing in (11)and thus also in the weight Z. By a redefinition UU


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U, the integration over U then reduces to a factor of 1and one is back at (3). (11) then leads to the contributionof real emissions to the evolution equation

(Y Z[U])real=

s2

Kxzy Uabz

iaUx ibUy

Z[U] . (13)

Inserting virtual corrections by the requirement of real-virtual cancellation in absence of interaction, one recov-ers the JIMWLK evolution as stated in (4, 5). Exclusivequantities on the other hand will depend separately onU and U and require to keep both fields in Z along withmore complicated evolution equations.

Since exclusive quantities are characterized by specificrestrictions on the phase space of produced gluons thephysically most transparent derivation of a correspondingevolution equation is built on a systematic constructionof the contributing real emission amplitudes. The firstmodification clearly concerns the -correlator M used toimplement the phase space integrals. Diffractive disso-ciation, which corresponds to a rapidity gap on the side

of the target, requires a factor u(k) = (Yk Ygap) foreach final state gluon with momentum k . (The gap ra-pidity Ygap is assumed to lie in the resolved range.) Themajor change, however, results from the appearance ofadditional diagrams that disappear in the inclusive re-sult through complete real virtual cancellation. Whilefor JIMWLK it is sufficient to consider branching pro-cesses that occur before the interaction with the Lorentzcontracted target, exclusive observables like diffractivedissociation will receive contributions from reabsorptionand production in the final state, i.e. after the inter-action with the target as shown in Fig.1. Reabsorption

FIG. 1: Generic diagrams for exclusive processes with finalstate interactions. In the diagrams, rapidity of gluons in-creases both vertically, in the final state, and horizontally,with the distance of their emission vertex to the target: Toleading logarithmic accuracy, ordering in Y coincides withordering in z towards the interaction region. Consequently,emissions into the final state after the interaction do not it-erate: lines marked in the graph to the right are suppressed.

of a gluon after the interaction in the amplitude takesa form similar to a virtual correction in the JIMWLKcase, but contains the soft interaction with the target,i.e. a factor U per hard particle. Technically, the neces-sary diagrams can be constructed by introducing a threetime formalism in which we distinguish z = , = 0and = + as the times at which the initial hard parti-cles are created, the interaction takes place and the finalstate is formed respectively. The transition amplitude

from z = to + is then created in two steps: weuse a shower operator to create gluons before the interac-tion but anticipate that some of them directly reach thefinal state while others will be reabsorbed after the inter-action. In order to also generate the final state contribu-tions with a shower operator, we introduce an auxiliaryGaussian noise with the same average and correla-tor as in (8) and (9). Furthermore we artificially split the

U factors of the interaction region into two Wilson linesW and V according to U = W V. (One may think ofthem as Wilson lines extending over the intervals [, 0]and [0, ], respectively, they will disappear in the finalresult.) We then obtain the full set of diagrams:

Uf[, ]Ui[] (14)

= Uf[, ]

|

|

|

|

|

|

|

|

|

|

|

|

|

|

|

|

=

|

|

|

|

|

|

|

|

where the sum is over the number of gluons and allowedinsertions. The dashed line through the interaction re-gion represents the auxiliary split of the Wilson lines intoW and V with accompanying factors. The shower op-erators are given by

Ui[, ] = PY2 exp

i

dY1dY2 (Y1Y2)J

ixz

(WabY2,zb,iY2,z

+ (W V)abY2,zb,iY2,z

)iaWY1x

(15a)

Uf[, ] = PY2 exp

i

dY1dY2 (Y1Y2)J

ixz

(VabY2,zb,iY2,z

+ a,iY2,z)iaVY

1,y . (15b)

Eventually combining the above expression for the ampli-tude with the corresponding expression for the complexconjugate amplitude and differentiating w.r.t. Y yieldsall real emission contributions to the evolution Hamil-tonian as well as the interacting virtual ones. One stillmisses virtual lines that do not cross the interaction re-gions. These are again reconstructed on the level of theevolution equation. We obtain the full Hamiltonian:

H = u(k)Hr + Hv + Hv (16)

where the real gluonic corrections are produced by

Hr = s2

Kxzy

Uabz iaUxibUy + Uabz iaUxi

bUy

+ (UU)abz iaUx

ibUy

+ iaUxiaUy

.

The remaining terms correspond to virtual corrections inamplitude and complex conjugate amplitude respectively

Hv=s22

Kxzy(iaUx

iaUy+iaUx

iaUy+2Uabz

iaUxibUy

)

Hv=s22

Kxzy(iaUx

iaUy

+iaUxiaUy

+2Uabz

iaUx

ibUy

) .


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(The last terms in these expressions are the interact-ing parts.) The evolution equation parallels (4), withZ replaced by Z[U, U]. Note that Hv and Hv taken in-dividually have the form of the JIMWLK-Hamiltonian:Hv is the evolution Hamiltonian for the dipole opera-tor of the forward amplitude N0

xy= tr[1 UxU

y

]/Nc,which, via the optical theorem, determines the evolu-tion of the total cross-section. Real contributions only

occur outside the gap, as mandated by the factor u(k).If we remove that restriction by setting u(k) = 1, weexpect complete cancellation of final state contributionsand again a reduction to JIMWLK. Indeed, setting u(k)to 1 and acting with (16) on the dipole operator NF

xy=

tr[1 (UU)x(UU)y]/Nc (which depends only on prod-

ucts (UU)()), we find that the evolution Hamiltonian(16) reduces to the JIMWLK-Hamiltonian for Wilsonlines (UU). The average over NF

xythen is written in

terms of Z[U, U] = Z[UU] with evolution according to

Y Z[UU] = H[UU]JIMWLKZ[UU

] , (17)

cancellation is complete.The operators that replace tr[1 UxUy]/Nc (i.e. N

F)in (1) for diffractive dissociation of a photon are differentin- and outside the gap, the structures closely resemblethe factorized results of [3]. In the gap, no colored objectenters the final state, thus the initial qq, if in the gap,interacts with the target and emerges in a singlet state:here ND

xy, the operator for cross sections with rapidity

gaps larger than Ygap, is a product of two traces:

NDxy

= N0xy

N0yx

=tr(1 UxUy)

Nc

tr(1 UxUy)

Nc, (18)

it takes the form of the square of two elastic dipoleoperators corresponding to the two amplitude factors.Here, evolution of amplitude and complex conjugate arecompletely uncorrelated ([Hv, Hv ] = 0, Hr is absent):

ZY [U, U] = e(Hv+Hv)(YY0)ZY0 [U, U] , (19)

and one may factorize ZY [U, U] ZY [U]ZY [U] unlessthe initial condition contradicts this.

For Y > Ygap, production of gluons in the final state isallowed and the qq-pair can appear also in an octet state.Adding the octet part to (18) removes one trace:

N

D

xy = tr[

1 U

yUx

1 U

x

Uy

]/Nc

= N0xy

+ N0yx

NFxy

. (20)

The second line exposes further structure: With the ini-tial conditions on evolution for ND imposed by (18), wefind that NF acquires the interpretation of the crosssection of events with rapidity gaps smaller than Ygap.Following our previous reasoning we conclude that theaverage over the three operators in the second line of(20) can be described by using functionals Z[U], Z[U]

and Z[UU], with their respective evolution given by aJIMWLK-Hamiltonian. Even if additional structure inthe initial conditions does not prevent these simplifica-tions, initial conditions for the individual terms are dif-ferent from each other (c.f. (18)) and the inclusive case.

The relation to the results of Kovchegov and Levin [3]parallels the reduction step from JIMWLK to BK: Thereone observes that JIMWLK evolution of Sxy[U] = 1

Nxy = tr(UxUy)/Nc takes the simple form

YSxyY =sNc22

d2zKxzySxzSzy SxyY , (21)

where Kxzy =(xy)2

(xz)2(zy)2 . Factorizing SxzSzy

SxzSzy truncates the infinite Balitsky hierarchy andleaves us with the BK equation. For Y > Ygap where

NDxy

= 1 Sxy[U] Syx[U] + Sxy[UU] and each

Sxy[..] obeys (21), the same reasoning leads to the evo-lution equation presented as Eq. (9) in [3]. (18) and(19) imply the required initial condition ND

xy(Ygap) =

N0xy2

(Ygap) for evolution above the gap.To summarize: We have developed a method whichallows to generalize the JIMWLK approach to a largeclass of exclusive observables, by simply adapting thephase space constraints. We have worked out the ex-ample of diffractive dissociation. For all generalizationsit is crucial to start from IR safe observables, otherwisereconstruction of virtual contribution via real virtual can-cellations must fail.

This work was supported in part by BMBF.

[1] for a recent review see H. Weigert, Prog. Part. Nucl. Phys.55 (2005), 461.

[2] J. Jalilian-Marian, A. Kovner, L.D. McLerran, and H.Weigert, Phys. Rev. D55(1997) 5414;J. Jalilian-Marian, A. Kovner, A. Leonidov, and H.Weigert, Nucl. Phys. B504 (1997) 415; Phys. Rev. D59(1999) 014014; Phys. Rev. D59 (1999) 034007, [Erratum-ibid. D59, 099903 (1999)];J. Jalilian-Marian, A. Kovner, and H. Weigert, Phys. Rev.D59 (1999) 014015;A. Kovner, J.G. Milhano, and H. Weigert, Phys. Rev. D62(2000) 114005;H. Weigert, Nucl. Phys. A703, 823 (2002);E. Iancu, A. Leonidov, and L.D. McLerran, Nucl. Phys.

A692 (2001) 583.E. Ferreiro, E. Iancu, A. Leonidov, and L.D. McLerran,Nucl. Phys. A703 (2002) 489;

[3] Y. Kovchegov and E. Levin Nucl. Phys. B577 (2000), 221.[4] I. Balitsky, Nucl. Phys. B463 (1996), 99.[5] Y. V. Kovchegov, Phys. Rev. D61 (2000), 074018.[6] A. Banfi, G. Marchesini, and G. Smye, JHEP 0208 (2002)

006; H. Weigert, Nucl. Phys. B685 (2004) 321.[7] Y. V. Kovchegov and L. D. McLerran, Phys. Rev. D60,

054025 (1999) [Erratum-ibid. D62, 019901 (2000)]; A.Kovner, U. Wiedemann Phys., Rev. D64 (2001) 114002.

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Martin Hentschinski, Heribert Weigert, and Andreas Schafer-Extension of the Color Glass Condensate Approach to Diffractive Reactions