C. BFKL Evolution & The Small x Problem

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1 C. BFKL Evolution & The Small x Problem Resum all terms ( α s ln 1 x) n, n 1, in xg(x, Q 2 ), even if non accompanied by powers of ln Q 2. Local in x, butnon localink Exponential increase with Y ln(1/x) ( rapidity ) Y ln 1 x ln s Q 2 dy = dx x dk zdb z xg(x, Q 2 ) dn dy (x, Q2 ) = Q 2 d 2 k dn dy d 2 k =#ofgluonsperunitrapidityandlocalized within a transverse size x 1/Q. N(Y ): # of gluons produced after a rapidity evolution Y p p k z 1 1 =x p p x 1 x << x 2 <<1 1 x << 1 k z =xp k x<< x z 1 =xp <<1 A 0 α s ln 1 x 1 n! x n << x << x n-1 x n ( αs ln 1 x) n N(Y ) A 0 n 0 n +1( αs Y ) n e ωα s Y = 1 n! x ωα s

2 Amorephysicalargument Gluons in the BFKL cascade are coherent in time. The lifetimes of the virtual gluons are strongly ordered along the cascade: t i 2 x i P k 2 i t 1 t 1... t n t All the gluons with x x are frozen over the natural time scale t for dynamics at x. The last gluon is emitted coherently off the global color charge of the previously emitted N(Y )gluons Random distribution of fast (x x) gluons Random color charge Q a = i Qa i Q a =0, Q 2 Y ( N ) 2 Q a i Y i=1 g 2 N c N(Y ) Indeed: for one gluon (gt a )(gt a )=g 2 N c The color charges of the N(Y )gluonssumup incoherently. Assumption: After being emitted, gluons do not interact with each other.

3 Probability for emitting a new gluon when Y Y +dy : dp (Y ) g 2 N c N(Y )dy ωα s N(Y )dy The average # of gluons at rapidity Y +dy : N(Y +dy )= [ 1+N(Y ) ] dp (Y )+N(Y ) [ 1 dp (Y ) ] = dn dy = dp dy = ωα sn(y )= N(Y ) e ωα sy Unstable growth of the gluon distribution! The radiated gluons act as sources for the emission of new gluons. BFKL equation provides the value of ω : ω =4ln2(N c /π) 2.65 together with the k spectrum of the emitted gluons. = The first problem of BFKL evolution at small x : Too rapid rise of total cross sections with 1/x (or s) 1) Unacceptable phenomenology 2) Conceptual difficulties

4 1) Phenomenological difficulties of BFKL How to compute F 2 when α s ln 1 x > ln Q2? Collinear factorization does not apply beyond DLA. New factorization schemes will be introduced later: i) k factorization ; ii) dipole picture In any case, at small x, F 2 is driven by the gluon distribution, so BFKL generically implies : F 2 (x.q 2 ) 1/x ωα s with ωα s 0.5 forα s =0.2 The data do rise indeed, but much slower! Parametrization : F 2 (x, Q 2 ) x λ eff (Q 2), x<0.01 HERA data: λ eff (Q 2 )fallsfrom0.35 to 0.1 when Q 2 falls from 200 GeV 2 to 1 GeV 2. Similar conclusions from other high energy experiments (γ γ scattering at L3, OPAL): The BFKL intercept α P 1 ωα s is far too large! Next to leading order (NLO) corrections to BFKL could reduce α P significantly. NLO BFKL become available recently: (Fadin, Lipatov (98); Ciafaloni, Camici (98)) and turned out to be larger then the leading order! ω NLO = ω LO ( ᾱs ) = 0.16 for α s =0.2

5 Resummations of higher order effects have been proposed to cure this lack of convergence problem : Salam, 98; Ciafaloni, Colferai, 99; Brodsky, Fadin, Kim, Lipatov, Pivovarov, 99; Schmidt, 99; Forshaw, Ross, Sabio Vera, 99 E.g.: The collinear resummation by Salam, Ciafaloni, and Colferai = sensible results for ω NLO : 0.35 Q 2 [GeV 2 ] LL 0.05 s c 0 NLL s (MSbar) From G. Salam, hep-ph/ Applications of NLO BFKL to phenomenology. Currently, a very active, and promising, field: γ γ scattering: encouraging results DIS:recentprogress,butsomeproblemsremain to be solved [Bartels, Colferai, Gieseke, Kyrieleis, Qiao]

6 Figure 1: H1 and ZEUS data on the F 2 structure function shown in three bins of Q 2 as a function of x. The steep rise of the structure function at low x is clearly apparent. From the review paper Lectures on HERA physics, by B. Foster, EPJdirect A1, 1 11 (2003); hep-ex/

7 H1 Collaboration H1 Collaboration Figure 2: The coefficients c(q 2 )andλ(q 2 )fromfitsof the form F 2 (x, Q 2 )=c(q 2 )x λ(q2) for x<0.01 H1prelim , T. Lastovicka, talk presented at DIS 2002, April 2002 & H1 contribution to EPS2003, Aachen, July 2003

8 50 s =91 GeV Data BFKL one-gluon FIT L3 (Y) [nb] s =183 GeV s =189 GeV Preliminary Y= ln(w / Q 1 Q 2 ) Figure 3: L3 results for the two photon cross section σ γ γ showing a strong deviation from LO BFKL. The continuous line is a BFKL like fit with α P 1leftasa free parameter. One finds: α P for s =91 GeV and α P for s =183GeV. From the review paper by Donnachie, J. Phys. G26: , 2000 hep-ph/

9 Figure 4: Virtual gamma-gamma total cross section by the NLO BFKL Pomeron vs L3 Collaboration data at energy 183 GeV of e + e collisions. Solid curves: NLO BFKL. Dashed: LO BFKL. Dotted: LO contribution. Two different curves are for two different choices of the Regge scale: s 0 = Q 2 /2, s 0 =2Q 2 From Kim, Lipatov, Pivovarov, hep-ph/ , presented to the VIIIth Blois Workshop, 1999

10 2) Unitarity violation by BFKL evolution σ γ p(x, Q 2 )= 4π2 α em Q 2 F 2 (x, Q 2 ) 1 x ωα s sωα s But hadronic crosss-sections cannot grow like a power of s in the high energy limit s! (A) Froissart bound: σ tot (s) σ 0 ln 2 s as s. Consequence of very general principles (unitarity, analitycity, crossing) +short rangeness(mass gap) In QCD one expects: σ 0 1/m 2 π. DIS: γ is a virtual state, not a hadron! Still: γ is a superposition of hadronic states = σ γ p(x, Q 2 ) σ 0 ln 3 1 x as x 0 Clearly, BFKL (LO or NLO) violates this condition. (B) Unitarity bound on the S matrix : SS =1 = S(x, b) 1foranyimpactparameterb. This condition too is violated by BFKL (see below). Is this a problem of the BFKL evolution of xg(x, Q 2 ), or of the calculation of F 2,orboth? One expects multiple scattering to restore unitarity in the S matrix. One may imagine that F 2 unitarizes, while xg(x, Q 2 )keepsgrowingasapowerof1/x.

11 The BFKL equation The unintegrated gluon distribution (with k k ): xg(x, Q 2 ) Q 2 dk 2 k 2 f(x, k2 ), f(x, k 2 )= xg(x, k2 ) ln k 2 BFKL equation (Y =ln(1/x) andᾱ s = α s N c /π) f(y,k 2 ) d 2 p k 2 { =ᾱ s Y π p 2 (k p) 2 f(y,p 2 ) 1 } }{{} 2 f(y,k2 ) }{{} real virtual Real :realgluonemissioninone stepevolution Virtual :vertexandself-energycorrections The virtual term compensates the infrared divergence of the real term at k = p. DLA limit comes out as expected: for k 2 p 2 f(y,k 2 ) Y k 2 dp 2 ᾱ s p 2 f(y,p2 )

12 The Gluon distribution of one quark Recall: gluon radiation by a fast quark ( bremsstrahlung ) (p-k) µ = ((1-z)p, -k,(1-z)p ) µ p =(p, 0,p) k µ =(zp, k, zp) dp = α sc F dk 2 1+(1 x) 2 2π k 2 dx x where C F t a t a =(Nc 2 1)/N c =4/3 One can identify this with the spectrum of the emitted gluons : dn dxdk 2 dp dxdk 2 α sc F π 1 1 k 2 x for x 1 Hence, the integrated gluon distribution generated by a single quark : f q (x, k 2 ) xg(x, k2 ) ln k 2 = k 2 x dn dxdk 2 = α sc F π The (integrated) gluon distribution of one quark: Q 2 xg q (x, Q 2 dk 2 α s C F )= k 2 = α sc F ln Q2 π π Λ 2 Λ=infraredcutoff The simplest proton : xg p (x, Q 2 ) 3 xg q (x, Q 2 )

13 The solution to BFKL equation for α s Y ln k 2 ( ) k f(y,k 2 2 1/2 { ( ) ) e ωᾱsy exp ln2 k 2 /k0 2 } 2βᾱ s Y k 2 0 where: ω =4ln2 2.77, β Exponential rise with Y. BFKL anomalous dimension 1/2 : f k 2 In LO perturbation th., f is independent of k. DGLAP evolution: f is a function of ln k 2. BFKL: strong violation of Bjorken scaling: F 2 Q 2 Diffusive behaviour in the variable ln k 2 with diffusion time Y. The evolution broadens the ln k 2 distribution of f. = New problem of BFKL evolution at small x : Infrared diffusion towards non perturbative momenta Even if the external momentum k is hard ( Λ 2 QCD ), the non-locality of the evolution makes it that, with increasing Y, f(y,k 2 )receivesanincreasinglylarge contribution from non perturbative p. Problems with the introduction of the running coupling α s (k 2 ) In practice: use an infrared cutoff k 2 c

14 k (k,y) BFKL 1.5 Kovchegov k [GeV] Figure 5: The functions kφ(k, Y )constructedfromsolutions to the BFKL and the Balitsky-Kovchegov equations for different values of the evolution parameter Y =ln(1/x) ranging from 1 to 10. The coupling constant α s =0.2. kφ(k, Y ) f(y,k)/k; kφ(k, Y 0 =0) = δ ( ln k 2 /k0 2 ) From K. Golec-Biernat, L. Motyka, A. M. Stasto, Phys Rev D65 (2002) ; hep-ph/

15 III. Unitarity vs. Saturation in DIS A. The idea of saturation Gribov, Levin and Ryskin, 83; Mueller and Qiu, 86 So far, the emitted gluons were assumed not to interact with each other Gluons within the same BFKL cascade are well separated in rapidity, so they cannot interact Gluons with similar rapidities but from different cascades interact with a cross section α s At small x, the gluon density is large,which enhances gluon recombination When RECOMBINATION = RADIATION = SATURATION What is the recombination probability? In order to interact, the small x gluons must overlap in transverse projection. Indeed, since λ z 1 xp 1 P,theirlongitudinal overlap is then automatic. xg(x, Q 2 )/πr 2 =#ofgluons(x, Q 2 )perunitarea.

16 Atypicalgg cross-section: σ gg (Q 2 ) α s /Q 2 The probability that a given gluon interacts with all other gluons with the same (x, Q 2 ): Γ(x, Q 2 ) α s(q 2 ) xg(x, Q 2 ) Q 2 πr 2 i) O(α s ), but amplified by xg(x, Q 2 ) x ω. ii) Suppressed by 1/Q 2 : highertwist. Γ(x, Q 2 ) 1forQ 2 Q 2 s (x) suchthat: Q 2 s (x) α s xg(x, Q 2 s (x)) πr 2 x λ Q s (x) Λ QCD provided x is small enough! = Ahardintrinsicscalewhichmaysuppress infrared diffusion and ensure weak coupling : α s ( Q 2 s (x) ) 1 for x 1 Q s (x) = saturationmomentum Fork Q s (x) :strongnon lineareffectswhich should saturate the gluon occupation number: n g (2π) 3 2 (N 2 c 1) dn dy d 2 k d 2 b 1 α s Fork Q s (x) :usuallinearevolution. n g 1, but rapidly increasing with 1/x and 1/k

17 With decreasing x, gluonsareproducedmostlyat large momenta k >Q s (x), andthuscannotbe seen by a probe (γ )withq 2 <Q s (x) = unitarization of DIS! n g 1 s f(y,k) k 2 Y 2 > Y 1 QCD Q ( Y 1 ) s s Y 2 Q( ) Y 1 k Gluon saturation is a natural solution to the small x problems of the linear evolution equations. How to compute in this non linear regime? Large occupation numbers Strong (A 1/g) classical fields. (cf. Introduction) Classical effective theory for the small x gluons

18 AcartoonofDISinthepresenceofnon lineareffects P Non linearities manifest themselves is: thegluondistribution,anditsevolutionwith1/x theproductionofthelastquark Strategy : 1) Factorize out the quark production and its subsequent interaction with γ (specific to DIS) Dipole Picture + Eikonal Approximation A. Mueller; Nikolaev, Zakharov; Kovchegov Buchmüller, Hebecker; Balitsky 2) Construct an effective theory for gluon correlations in the hadron wavefunction at small x Color Glass Condensate McLerran, Venugopalan: MV model (classical model for a large nucleus A 1) E.I., Jalilian-Marian, Kovner, Leonidov, McLerran, Weigert: QCD at small x

19 B. Light cone kinematics & quantization Afastmovingparticleinthepositivez direction: p z m, v c : z = t, x, y =const. Light cone coordinates: x (t + z), x 1 2 (t z), x =(x, y) Particle at rest at x =0with time x + For a particle moving in the negative z direction (z = t), the roles of x + and x get interchanged. 4 vector: x µ =(x +,x,x ) Measure: d 4 x =dx + dx d 2 x Metric: k x = k x + + k + x k x k : LCenergy ;k + : LClongitudinalmomentum on shell excitation : k 2 = m 2 = k = m2 + k 2 2k + Proton s IMF: P z P M = P µ δ µ+ P + P + 2P, P = M 2 /2P + P + Boost invariant longitudinal momentum fraction: ( x k+ P + k ) z in IMF P z and rapidity: Y =ln(1/x) =ln(p + /k + )

20 Gluon Distribution xg(x, Q 2 ) Q 2 d 2 k dn dy d 2 k LC quantization of the SU(3) gauge fields LC gauge: A + a =0( temporal gauge A 0 a =0) Fock space gluon number density: dn d 3 k dn dk + d 2 k = λ,c a λ,c ( k)a λ,c ( k) [ k (k +, k)] One finds (with F a +i(x) = + A i a (x) inlcgauge): dn dy d 2 k = 1 F +i 4π 3 a (x+, k)f a +i (x+, k) where :averageoverhadronwavefunction. This is a gauge invariant quantity, but coincides with the Fock space gluon number only in LC gauge. In some arbitrary gauge, the r.h.s. is replaced by O γ,withthegauge invariantoperator: O γ ( x, y) Tr { F i+ ( x) U γ ( x, y) F i+ ( y) U γ ( y, x) } where x =(x, x) andthe Wilson line: { } U γ ( x, y) =Pexp ig d z A a ( z)t a γ

21 t x - x + z x (-, 8 x ) (x, x ) x- (-, 8 y ) (y, y ) The path γ used for the evaluation of the gauge-invariant operator O γ. Gauge transformations: h(x) SU(3), hh =1 A µ (x) A µ a(x)t a h(x)a µ (x)h (x) + i g h(x) µ h (x) U γ (x, y) h(x) U γ (x, y) h (y)

22 C. DIS in the Dipole Frame How to compute F 2 in the presence of higher twist effects? The struck quark: the last emitted parton in an otherwise purely gluonic cascade It is convenient/possible to disentangle this last quark from the gluon evolution via a Lorentz boost : Give γ enough energy to fluctuate into a q q pair long time before the collision:. boost γ = c 0 γ 0 + c 1 q q 0 + (c 1 α em ) q q 0 : ColorDipole (colorsinglestate); To O(α em ), DIS is determined by the dipole scattering off the color fields in the hadron.

23 Dipole frame : P µ (P, 0,P) & q µ =( q 2 Q 2, 0, q) with q Q but such that α s ln(q/q) 1 i) The q q fluctuation has a long lifetime: τ q q ( E q q E γ ) 1 τdis R/γ 1/P = the dipole transverse size r = x y is frozen (unchanged by the dipole hadron interaction). ii) No gluon (QCD evolution) in the q q wavefunction. Most of the total energy, and all the evolution, are still carried by the proton! α s ln(1/x) α s ln(p/q) > 1 Small x DIS in the Dipole Frame : Dissociation of γ into a q q pair followed (long time after) by the interaction between a bare color dipole and a highly evolved hadron. σ γ p(x, Q 2 )=σ T + σ L 1 σ T,L (x, Q 2 )= dz d 2 r ΨT,L (z,r; Q 2 ) 2 σ dipole (x, r) 0 Ψ 2 O(α em ):LC wavefunction for the dissociation of γ into a q q pair of size r and a longitudinal fraction of the quark equal to z All the QCD dynamics is encoded in σ dipole.

24 How to compute σ dipole? b P r x y r = x-y b = (x+y)/2 Eikonal approximation: Relative motion is very fast (s qp k 2 ) straightlinetrajectories(x,y =fixed) couplingtoa + a alone: j q µ A µ A + (since q and q propagate in the negative z direction: j q µ δ µ ) (Use a gauge different from LC gauge A + =0!) σ dipole (x, r) =2 d 2 b ( ) 1 Re S(x, r, b) S(x,r, b) :S matrix at impact parameter b. S 1(unitaritybound,fromSS =1) S 1:weakscattering(transparency); S 1:completabsorbtion(blackness)

25 The S matrix in the eikonal approximation: i) Single quark in the background of the field A + a : { S q (x) =Pexp ig dx A + a (x, x)t a} V (x) + A + A ii) Color dipole in the fluctuating field of a hadron: S Y (x, y) = 1 tr V (x) V (y) S Y (r, b) N c Schematically : S = φ f φ i = φ i U(, ) φ i { U(, ) =Texp i dt } d 3 x L int (t, x) x - Here: time = x, x =(x +, b), L int = j µ a Aa µ with jµ a = gta δ µ δ(x + )δ(b x) (1/N c )tr:averageovercolorfortheq q pair V, V SU(3) : Wilson lines Color precession of the quark (or antiquark) after crossing the field. V, V :AllordersingA + = Multiple scattering Y

26 Weak fields (ga + 1) = Perturbative expansion Appropriate at not so high densities (not so small x). Lowest order in α s :Single scatteringapproximation ( ) S Y (x, y) 1 g2 2 A + a 4N (x) A+ a (y) c Y A + A + gluon distribution. Exercice i) Show that to O(α s ): σ dipole (x, r) 4π N c α s d 2 k k 4 f(x, k2 ) (1 e ik r) ii) If f(x, k 2 )isslowlyvaryingink 2,showthat: σ dipole (x, r) π2 N c α s r 2 xg(x, 1/r 2 ) The weak field/coupling expansion to a given order violates the unitarity bound. When ga + 1, any number of collisions is important, and unitarity should be restored. When ga + 1, non lineareffectsinthegluon distribution become important too. Unitarization of dipole scattering & Saturation of the gluon distribution: Two aspects of the same non linear physics! σ dipole (x, r) πr 2 when r 1/Q s (x)