Universality and tree structure of high energy QCD

0.2 MB·English

by  S. Munier

Universality and tree structure of high energy QCD
S. Munier∗
Centre de Physique Th´eorique, E´cole Polytechnique, 91128 Palaiseau Cedex, France†
R. Peschanski‡
Service de Physique Th´eorique, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France§
Using non-trivial mathematical properties of a class of nonlinear evolution equations, we obtain
the universal terms in the asymptotic expansion in rapidity of the saturation scale and of the
unintegrated gluon density from the Balitski˘ı-Kovchegov equation. These terms are independent
of the initial conditions and of the details of the equation. The last subasymptotic terms are new
results and complete the list of all possible universal contributions. Universality is interpreted in a
generalqualitativepictureof highenergy scattering, inwhich ascatteringprocess correspondstoa
4 treestructure probed by a given source.
0
0
2 1. Introduction the BKequation(1) is the prototype ofsaturationequa-
tions.
n
a Theenergydependenceofhardhadroniccrosssections
J is encoded in the evolution of the parton distributions However, besides the mathematical challenge posed
7 with the rapidity Y. If the relevant scale Q is large by the nonlinear character of saturation equations,
2 there is up to now no fully consistent way to include
enough, like in the case of deep-inelastic scattering, for
saturation corrections in the perturbative expansion of
which Q is the virtuality of the photon, the latter can
1
QCD. Furthermore, initial conditions are required at
v be computed in perturbative QCD. The Balitski˘ı-Fadin-
some low rapidity Y , for all values of k: it is a priori
5 Kuraev-Lipatov(BFKL) equation[1] resums the leading 0
1 Feynman diagrams, in the form of a linear integrodiffer- not clear how dependent on them the solutionof Eq. (1)
2 ential evolution equation for the gluon density. Its solu- is, in particular for small k ΛQCD, which lies beyond
∼
1 the known perturbative domain. Such sensitivity on
tion exhibits an exponential growth with Y. At larger
0
the infrared has already been a major drawback to the
Y, maybe already attainable at present colliders [2], the
4
BFKL equation [9]. One purpose of this Letter is to
0 BFKL equation breaks down because finite parton den-
/ sity or saturation effects become sizeable [3, 4]. It is explaininwhatrespectthesolutionstotheBKequation
h
are independent of the details of both the nonlinear
p replaced by nonlinear evolution equations which aim at
evolution equation and of the initial conditions. Let us
- describing such effects [5, 6, 7]. The appropriate equa-
p call this property universality.
tionsdependonthephysicalobservablesandonthelevel
e
h of approximation of QCD. They could also include non-
v: perturbative contributions [8]. Our aim is to show that 2. Universal properties of the solutions of the BK
minimalassumptionsonthestructureofsaturationequa-
i equation
X tions leadto quite general,i.e. “universal”,propertiesof
r the solutions. As a case study, we will consider the Bal-
a
itski˘ı-Kovchegov(BK) equation [5, 7]. We are going to use general methods for solving non-
In the large Nc limit, at fixed QCD coupling α¯ = linear evolution equations [10, 11]. The mathematical
αsNc/π,andforahomogeneousnucleartarget,themea- results that will be relevant for the BK equation require
sured gluon density (k,Y) at transverse momentum k that three main conditions be fulfilled [11]. (i) = 0
N N
obeys the BK equation, which reads should be a linearly unstable fixed point, i.e. any small
fluctuationof growstoinfinityfromthelinearizedevo-
∂ =α¯χ( ∂ ) α¯ 2 . (1) N
Y L lution equation; (ii) the effect of the nonlinearity is to
N − N − N
damp this growth when (1); (iii) the initial con-
∂L standsforthepartialderivativewithrespecttologk2 dition must be “sufficientNly”∼stOeep at largek. Properties
and the function χ is the BFKL kernel χ(γ) = 2ψ(1)
− (i) and (ii) stem from the parton model in general and
ψ(γ) ψ(1 γ). The nonlinear term in the r.h.s. corre-
− − are verified by the BK equation in particular. In the
spondstothesaturationterm. Foritsrelativesimplicity,
case of the BK equation, as noticed in Refs. [12, 13],
the constraint (iii) means (k,Y ) 1/k2γc, with γ =
0 c
N ≪
0.6275 . Color transparency of perturbative QCD im-
···
plies (k,Y ) 1/k2 at large k, which fulfills this con-
†UMR7644,unit´emixtederechercheduCNRS. N 0 ∼
§URA2306,unit´ederechercheassoci´eeauCNRS. dition.
∗Electronicaddress: [email protected]
‡Electronicaddress: [email protected] Theclassofequationsdefinedbythelatterconstraints
2
admit traveling wave solutions1, i.e. at large Y, is a
N
uniformly translating front:
Y
∂
N(k,Y)∼N∞ logk2−logQ2s(Y) . (2) 2g Q/s
The mathematical treatm(cid:0)ent of Eq. (1) con(cid:1)sists in con- ∂ lo
sidering asymptotic expansions in Y of both the satura-
tionscale(relatedtothetranslationofthetravelingwave
front with rapidity [12]) and of the gluon distribution
N
(which is the frontprofile [13]). It leads to a hierarchical
systemofnestedordinarydifferentialequations. Ateach
order of the expansion, the integration constants are de-
terminedbyamatchingprocedure[11]. Thecorrespond- O(1)
inggeneralequationscanbestraightforwardlytranslated O(1/Y)
to the case of the BK equation.
O(1/Y3/2)
Let us first discuss the saturation scale Q (Y). In
s
terms of the parameters of the BK equation, the sys-
tematic expansion2 of Q (Y) in Y reads
s
Y
χ(γ ) 3
logQ2(Y)=α¯ c Y logY
s γc − 2γc FIG. 1: The effective dependence ∂logQ2(Y)/∂Y of the sat-
s
3 2π 1 uration scale upon the rapidity Y. α¯ was set to 0.2. The
+ (1/Y) . (3) truncations of the asymptotic expansion Eq. (3) to order 1,
− γc2sα¯χ′′(γc)√Y O 1/Y and 1/Y3/2 are shown.
γ is the solution of the implicit equation χ(γ ) =
c c
γ χ′(γ ), see Ref. [3]. The successive orders contribut-
c c
ing to ∂logQ2(Y)/∂Y are displayed in Fig. 1. is expanded about its asymptotic shape , see
s N N∞
Afewcommentsareinorder. ThefirstterminEq.(3) Eq. (2). In that region, 1 and the full nonlinear
N ∼
is the dominant evolution term [16]. The second term equation must be considered. In the “leading edge” de-
[17] reflects the delay in the formation of the front [13]. fined by z 1, the wave front is expanded in powers of
∼
The third term is a new result for the solution of the 1/√Y: it is the transition region,where the front forms.
BK equation. These three terms are universal. They Going from the front interior to the leading edge, the
are the only possible universal terms in this asymptotic asymptotic front profile crosses over to [11]
∞
N
expansion. Indeed, one sees that by performing a shift
tmineortmdhisef,yriwnapghiidtlehitetyhinYeitl→ioawleYcsotn+odrYitd0ieoirnnst,Eeroqmn.es(3ga)er,testhnaaodttdaiatmffioeoncuatneltd1s/btYyo N(k,Y)=C1(cid:18)Q2sk(2Y)(cid:19)−γce−z2 ×(γclog[k2/Q2s(Y)]
the shift. +C + 3 2C + γcχ(3)(γc) z2
We now turn to the discussionof the properties of the 2 − 2 χ′′(γ )
(cid:18) c (cid:19)
front. The whole method to get solutions relies on the 2γ χ(3)(γ ) 1
matching between two different expansions of the wave − 3 cχ′′(γ )c + 32F2 1,1;52,3;z2 z4
front above the saturation scale: the so-called “front (cid:18) c (cid:19)
(cid:2) (cid:3)
interior” and “leading edge” expansions [11]. In the
+6√π 1 F 1,3;z2 z+ (1/√Y) (4)
“front interior”, the relevant small parameter is z = −1 1 −2 2 O )
log(k2/Q2(Y))/ 2α¯χ′′(γ )Y 1, i.e. one stands at (cid:0) (cid:2) (cid:3)(cid:1)
s c ≪
transverse momenta k near the saturation scale Qs(Y). where C1 and C2 are two integration constants and 1F1,
p
F are hypergeometric functions.
2 2
The first term can be identified to e−z2 near
∞
N ×
the front, and reproduces the known results [13, 16, 17]:
1 Rigorous mathematical results are available [14] for the Fisher induces so-called geometric scaling [18], while
∞
and Kolmogorov-Petrovsky-Piscounov [15] equation (related to Ne−z2 describes the diffusive formation of the front, and
the BK equation in the diffusive approximation [12]), and have
inducesaspecificpatternofgeometricscalingviolations.
beenextended tomoregeneralcases[11].
2 NotethatthescaleofQs(Y)isincludedinthedefinition. Indeed, All other terms appearing in the expansion (4) are
the saturation scale Qs(Y) can be defined, e.g., in such a way new terms that represent the (1/√Y) corrections to
that N(Qs(Y),Y)=N0, where N0 is a given constant. Qs(Y) e−z2. Note that they dOepend also on the third
is the position of the traveling wave front. All universal terms ∞
N ×
areindependent ofN0. derivative χ(3)(γc), at variance with the new term of
3
order1/√Y intheexpressionforthesaturationscale(3). probe becomes strong, and the probe can interact with
several partons simultaneously, see Fig. 3. This is the
saturationtransition, and d(Y) plays the role of the sat-
uration size 1/Q (Y).3 As d(Y) gets even smaller, the
s
partons inside the tree may interact and recombine. All
e
il these finite density or saturation effects generate nonlin-
f
o
r ear damping terms in the evolution equations that slow
p
t downtheevolution. Dependingonthephysicalsituation,
n
o the evolution of the interaction amplitude in this regime
r
d f is e.g. described by the BK equation, or by the Balit-
ce ski˘ı [5] and JIMWLK [6] equations which include more
u
d saturation effects.
e
r The structure described here complies with the three
abovementionedconditions(i), (ii), (iii)foranevolution
equationtoadmittravelingwavesolutions. Theprevious
discussionisgeneralandappliesto variousmodifications
of the saturation equations. Therefore, we expect the
universalresultspresentedheretobevalidingeneral,up
to the replacement of the relevant parameters related to
thebasictreeandwhichcharacterizethelinearevolution
log(k2/Q2) kernel.
s
FIG. 2: The reduced front profile (k,Y) (k2/Q2(Y))γc 4. Conclusion and discussion
N × s
in the leading edge . The curves are for Y = 10 (lower
curves) to Y = 50 (upper curves). α¯ = 0.2 and the con- We have pointed out that the structure of the QCD
stants in Eq. (4) were set to C1 = 1, C2 = 0. The lead- parton model is related to general cascading processes
ing edge expansion (4) is truncated at level (√Y) (dashed
O possessing specific mathematical properties. We have
lines) and (1) (solid lines). The reduced asymptotic front,
i.e. N∞(k,OY)×(k2/Q2s(Y))γc (see Eq. (2)), is the straight oabsytmainpetodtiacllexupnaivnesriosanlotfertmhessiantculuradtiniognnsecwaleoannesdionf tthhee
line.
unintegrated gluon distribution, see Eqs. (3,4). Univer-
salitymeansindependenceofthesolutionwithrespectto
3. Physical interpretation of universality
the initial conditions after a long enough rapidity evolu-
tionanddependenceofthesolutiononafewrelevantpa-
Let us discuss the physicalpicture underlying the uni-
rametersobtainedfromthekernelofthelinearevolution:
versalitypropertiesofageneralscatteringprocessinhigh
γ , χ(γ ), χ′′(γ ) for the saturation scale, plus χ(3)(γ )
c c c c
energy QCD, beyond the specific example of the BK
for the front. The results have been obtained from the
equation. The linear kernels of the saturation equations
BK equation, but are straightforwardly applicable to a
correspond to a developing tree of partons. The tree
wholeclassofsaturationequations. Thesaturationscale
structureisgeneratedbysuccessivebranchingofpartons
and the gluon density above the saturation scale keep
whentherapidityY increases,whichstemsfromthenon-
exactlythe samefunctionalformforthe universalterms.
abelian character of QCD. For example, the BFKL ker-
Finally, it is interesting to look for the concrete im-
nelisknowntocorrespondtoaspecifictreeofgluonsor,
printsofsaturationontheuniversalpartsofthesolution,
equivalently at large N , of cascading dipoles [19].
c
Eqs. (3,4). They appear in two places: the coefficient of
In the course of the parton branching,the partons get
the second term of the expansion of the saturation scale
more densely packed together, and thus the color field
(3 insteadof 1 for BFKL,see Eq.(3)), andthe logarith-
gets stronger. The strengh of the field can be qualita- 2 2
mic factor in in the expression for the asymptotic
tively characterized by the average transverse distance N∞
front(Eq.(4)), that isresponsibleforthe developinglin-
d(Y) between neighboring partons.
earshapeofthereducedfrontprofileintheleftmostpart
The scattering process corresponds to the interaction
ofthe leading edge, see Fig. 2. This factor appearsto be
of a source with the partons. As long as d(Y) is larger
a general dynamical property of absorbing walls, near
thantheresolution1/Qoftheprobe,thefieldseenbythe
probe is weak and thus the interaction merely “counts”
the partons. This is the dilute regime, in which the evo-
lution of the observable follows the exponential growth
3 This phenomenon is conventionally interpreted as the onset of
of the parton density with Y given by the BFKL equa-
theblackdiskduetothefillingofthephasespacebypartonsof
tion. Whend(Y)reaches1/Q,the colorfieldseenbythe size1/Q[3].
4
Y1 Y2 with running coupling. The latter falls in a different
Y
class of nonlinear equations, but it can be investigated
throughthesamegeneralmethods,showingthepowerof
x, y
the mathematical approach.
We thank Edmond Iancu and Dionysis Triantafyl-
lopoulos for their comments.
d(Y1 ) d(Y2 )
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