BFKL Pomeron in Nuclear Physics Andrey Tarasov

BFKL pomeron in nuclear physics Andrey Tarasov

St. Petersburg State University

1 Deep inelastic scattering

Deep inelastic regime: Q2 = q2 M 2 proton P k X k+q Q2 Bjorken variable: x = s + Q2 M 2 q electron l l’ This kinematic variables are ﬁxed

Fig. 5: Left: the DIS process. Right: the absorptionby of the initial virtual photon condition by a quark. ( l, P ) and l0

so these fluctuations are now well separated from the those of the vacuum (which have a lifetime They1/ ⇤determinein any frame, transverse since the vacuum area is boost 1 invariant)./Q and The lifetime (2) is much larger than ⇠ QCD longitudinalthe duration of amomentum typical collision process of the (see below); parton so, forxP the purpose of scattering, the hadronic fluctuations can be viewed as free, independent quanta. These quanta are the partons (a term coined by thatFeynman). is involved It then becomes in the possible scattering to factorize the cross–section (say, for a hadron–hadron collision) into the product of parton distribution functions (one for each hadron partaking in the collision), which describe the probability to find a parton with a given kinematics inside the hadronic wavefunction, and partonic cross–sections, which, as their name indicates, describe the collision between subsets of partons from the target and the projectile, respectively. If the momentum transferred in the collision is hard enough, the partonic cross–sections are computable in perturbation theory. The parton distributions are a priori non–perturbative, as they encode the information about the binding of the partons within the hadron. Yet, there is much that can be said about them within perturbation theory, as we shall explain. To that aim, one needs to better appreciate the role played by the resolution of a scattering process. In turn, this can be best explained on the example of a simpler process: the electron–proton deep inelastic

scattering (DIS). 2 The DIS process is illustrated in Fig. 5 (left): an electron with 4–momentum `µ scatters off the proton by exchanging a virtual photon (⇤) with 4–momentum qµ and emerges after scattering with 4–momentum `0 = ` q . The exchanged photon is space–like : µ µ µ 2 2 2 2 q =(` `0) = 2` `0 = 2E E (1 cos ✓ ) Q with Q > 0, (3) · ` `0 ``0 ⌘ with E = ` , E = ` , and ✓ = (`, ` ). The positive quantity Q2 is referred to as the ‘virtuality’. ` | | `0 | 0| ``0 \ 0 The deeply inelastic regime corresponds to Q2 M 2, since in that case the proton is generally broken by the scattering and its remnants emerge as a collection of other hadrons (denoted by X in Fig. 5). The (inclusive) DIS cross–section involves the sum over all the possible proton final states X for a given `0. A space–like probe is very useful since it is well localized in space and time and thus provides a snapshot of the hadron substructure on controlled, transverse and longitudinal, scales, as fixed by the kinematics. Specifically, we shall argue that, when the scattering is analyzed in the proton IMF, the virtual photon measures partons which are localized in the transverse plane within an area ⌃ 1/Q2 ⇠ and which carry a longitudinal momentum kz = xP , where x is the Bjorken variable :

Q2 Q2 x = , (4) ⌘ 2(P q) s + Q2 M 2 · where s (P + q)2 is the invariant energy squared of the photon+proton system. That is, the two ⌘ kinematical invariants Q2 and x, which are ﬁxed by the kinematics of the initial state (`,P) and of the scattered electron (` ), completely determine the transverse size ( 1/Q) and the longitudinal momentum 0 ⇠ fraction (x) of the parton that was involved in the scattering. This parton is necessarily a quark (or

10 Deep inelastic scattering

Emission of intermediate gluonq is q enhanced by large factor ln(1/x)

↵ 1, ↵ ln(1/x) 1 s ⌧ s ⇠ BFKL equation sums all ladder diagrams

2 @N(y, r1,r2) ↵¯ 2 r12 = d r3 2 2 N(y, rP1,r3)+N(y, r2,r3) N(y, r1,r2) P @y 2⇡ r13r23 Z ⇣ ⌘

H1 and ZEUS 1 Fig. 9: Parton evolution in perturbative QCD. The parton cascade on the right involves only gluons (at intermediate xf 2 2 Q = 10 GeV stages) and is a part of the BFKL resummation at small x (see the discussion around Eq. (22)). HERAPDF1.0 0.8 At large densities saturation is important. exp. uncert. model uncert. parametrization uncert. xuv One should take into account non-linear effects 0.6 gluon distribution xg (× 0.05) is known as the . The last estimate above follows from the uncertainty principle: partons with longitudinal momentum kz = xP are delocalized in z over a distance z 1/kz. Hence, 0.4 ' xdv the gluon distribution yields the number of gluons per unit of longitudinal phase–space, which is indeed xS (× 0.05) the right quantity for computing the occupation number. Note that gluons with x 1 extends in z 0.2 ⌧ over a distance z 1/xP which is much larger than the Lorentz contracted width of the hadron, ⇠ R/ 1/P . This shows that the image of an energetic hadron as a ‘pancake’, that would be strictly 10-4 10-3 10-2 10-1 1 ⇠ x correct if the hadron was a classical object, is in reality a bit naive: it applies for the valence quarks with x (1) (which carry most of the total energy), but not also for the small–x partons (which are the Left: the 1/x–evolution of the gluon, sea quark, and valence quark distributions for Q⇠2 =O 10 GeV2, as Fig. 11: most numerous, as we shall shortly see). measured at HERA (combined H1 and ZEUS analysis [4]). Note that the gluon and sea quark distributions have 2 been reduced by a factor of 20 to fit inside the figure. Right: the ‘phase–diagram’ for parton evolution in QCD; x Q (ii) When decreasing at a fixed , one emits mostly gluons which have smaller longitudinal3 2 each coloured blob represents a parton with transverse area x 1/Q and longitudinal momentum kz = xP . ? momentum fractions, but which occupy, roughly, the same transverse area as their parent gluons (see 2 ⇠ The straight line ln Qs(x)=Y is the saturation line, cf. Eq. (25), which separates the denseFig. and 11 dilute right). regimes. Then the gluon occupation number, Eq. (17), increases, showing that the gluonic system evolves towards increasing density. As we shall see, this evolution is quite fast and eventually leads to a This growth is indeed seen in the data: e.g., the HERA data for DIS confirmbreakdown that the proton of the picture of independent partons. wavefunction at x<0.01 is totally dominated by gluons (see Fig. 11 left). However, on physical grounds, such a rapid increase in the gluon distribution cannot go on for ever (that is, downIn to arbitrarily order to describe the small–x evolution, let us start with the gluon distribution generated by small values of x). Indeed, the BFKL equation is linear — it assumes that the radiateda single gluons valence do not quark. This can be inferred from the bremsstrahlung law in Eq. (16) (the emission interact with each other, like in the conventional parton picture. While such an assumptionprobability is perfectly is the same as the number of emitted gluons) and reads legitimate in the context of the Q2–evolution, which proceeds towards increasing diluteness, it eventually Y 2 breaks down in the context of the –evolution, which leads to a larger and larger gluon density. As long dN ↵ C Q dk2 ↵ C Q2 as the gluon occupation number (17) is small, n 1, the system is dilute and the mutual interactions g 2 s F s F ⌧ x (Q )= 2? = ln 2 , (19) of the gluons are negligible. When n (1), the gluons start overlapping, but their interactions are dx ⇡ ⇤2 k ⇡ ⇤ ⇠ O Z QCD QCD ! still weak, since suppressed by ↵ 1. The effect of these interactions becomes of order one only ? s ⌧ when n is as large as n (1/↵ ). When this happens, non–linear effects (to be shortly described) ⇠ O s where we have ignored the running of the coupling — formally, we are working to leading order (LO) in become important and stop the further growth of the gluon distribution. This phenomenon is known as pQCD where the coupling can be treated as fixed — and the ‘infrared’ cutoff ⇤QCD has been introduced gluon saturation [5–7]. An important consequence of it is to introduce a new transverse–momentum as a crude way to account for confinement: when confined inside a hadron, a parton has a minimum scale in the problem, the saturation momentum Qs(x), which is determined by Eq. (17) together with the virtuality of (⇤2 ). In Eq. (19) it is understood that x 1. In turn, the soft gluon emitted by condition that n 1/↵s : O QCD ⌧ ⇠ the valence quark can radiate an even softer gluon, which can radiate again and again, as illustrated in xg x, Q2(x) 2 2 1 2 s figure 10. Each emission is formally suppressed by a power of ↵ , but when the final value of x is tiny, n x, Q = Qs(x) = Qs(x) ↵s 2 . (23) s ⇠ ↵s ) ' R the smallness of the coupling constant can be compensated by the large available phase–space, of order Except for the factor ↵s, the r.h.s. of Eq. (23) is recognized as the density of gluonsln(1 per/x unit) per transverse gluon emission. This evolution leads to an increase in the number of gluons with x 1. 2 area, for gluons localized within an area ⌃ 1/Qs(x) set by the saturation scale. Gluons with k ⌧ ⇠ ?  Q (x) are at saturation: the corresponding occupation numbers are large, n 1/↵ , butFor do not a quantitativegrow estimate, consider the first such correction, that is, the two–gluon diagram in s ⇠ s anymore when further decreasing x. Gluons with k Qs(x) are still in a dilute regime:Fig. 10 the left: occupation the region in phase–space where the longitudinal momentum fraction x1 of the intermediate ? numbers are relatively small n 1/↵ , but rapidly increasing with 1/x via the BFKL evolution. The ⌧ s separation between the saturation (or dense, or CGC) regime and the dilute regime is provided by the saturation line in Fig. 11 right, to be further discussed below. 16 The microscopic interpretation of Eq. (23) can be understood with reference to Fig. 12 (left) : gluons which have similar values of x (and hence overlap in the longitudinal direction) and which occupy 2 a same area 1/Q in the transverse plane can recombine with each other, with a cross–section gg g 2 ⇠ ! ' ↵s/Q . After taking also this effect into account, the change in the gluon distribution in one step of the

18 Balitsky-Kovchegov equation

2 @N(y, r1,r2) ↵¯ 2 r12 = d r3 2 2 N(y, r1,r3)+N(y, r2,r3) N(y, r1,r2) @y 2⇡ r13r23 Z ⇣ ⌘ ↵ 1, ↵ ln(1/x) 1 s ⌧ s ⇠

2 @N(y, r1,r2) ↵¯ 2 r12 = d r3 2 2 N(y, r1,r3)+N(y, r2,r3) N(y, r1,r2) N(y, r1,r3)N(y, r3,r2)) @y 2⇡ r13r23 Z ⇣ ↵ 1, ↵ ln(1/x) 1, ↵2A1/3e!(0)y 1 γ∗ s ⌧ s ⇠ s ⇠ ↵2e!(0)y 1 ⌘ s ⌧ γ∗

Non-BK pomeron loops ↵2e!(0)y 1 ⌘ s ⇠

4 Small-x interactions in QCD There are some alternative approaches to describe scattering in QCD at small-x (high-energies): 1) Balitsky’s Wilson-line approach 2) JIMWLK approach, based on the idea of strong gluonic ﬁelds 3) Colour dipole model 4) RFT-like diagram technique of interacting BFKL pomerons

The identity of the approaches is not fully clear. But fortunately all basic results, like Balitsky-Kovchegov equation, are equivalent to each other.

5 Small-x interactions in QCD Equivalence of all this models imply that dynamic of interaction is based on the BFKL pomeron exchange

Gluons Impact factors Pomeron propagator Im (s, t) (k ,q) (k ,q) A = G d2k d2k 1 1 2 2 F (s, k ,k ,q) s (2⇡)4 1 2 k2(k q)2 1 2 Z 2 1

One can try to formulate an effective ﬁeld theory of the BFKL pomeron which is based on its general properties and does not deal directly with underlying gluonic dynamic.

6 Effective local pomeron ﬁeld theory

A predecessor of this theory, reggeon ﬁeld theory, was formulated a long time ago. = + + L L0 LI LE = †S + †( + †)+g⇢ L Interaction with a nucleon 1 2 S = (!@ @ )+↵0 + ✏ ⇢ = AT (b)(y) 2 y y rb The theory uses Lagrangian approach. One can use standard Feynman diagram calculations.

Classical equations of motion. S S =0, =0 (y, b) †(y, b)

@† 2 2 ↵0 † ✏† † = g⇢(y, b) @y rb gAT (b)e✏y †(y, b)= 1+gAT (b) 1 (e✏y 1) ✏ A. Schwimmer, Nucl. Phys. B 94 (1975) 445

7 Effective BFKL pomeron ﬁeld theory

Strong interaction at high energies is mediated by the exchange of BFKL pomerons BFKL pomerons

In the limit N c they split and fuse by triple pomeron!1 vertex

For hadron-nucleus scattering the relevant tree (fan) diagrams are summed by the Balitsky- Kovchegov (BK) evolution equation BFKL pomerons

8 Effective BFKL pomeron ﬁeld theory

We use effective non-local ﬁeld theory with = 0 + I + E L L L L

2 2 2 2 @ = d r d r † + H L0 1 2 r1r2 @y BFKL Z ⇣ ⌘ BFKL pomeron propagator

Triple pomeron vertex

interaction with the nuclear target

2 2 = d r d r †J LE 1 2 Z

2 2 2 2 2↵sNc d r1d r2d r3 = †(y, r ,r )†(y, r ,r )K (y, r ,r )+( †) LI ⇡ r2 r2 r2 1 2 2 3 31 3 1 $ Z 12 23 13

M. Braun, Phys. Lett. B 632 (2006) 297

9 Balitsky-Kovchegov equation Can be obtained it this approach as a solution of classical equations of motion S S =0, =0 (y, r1,r2) †(y, r1,r2)

@(y, r ,r ) ↵¯ r2 1 2 = d2r 12 (y, r ,r )+(y, r ,r ) (y, r ,r ) (y, r ,r )(y, r ,r ) @y 2⇡ 3 r2 r2 1 3 2 2 1 2 1 3 2 3 Z 13 23

BFKL pomerons

rapidity

2 BK approximation can be justiﬁed if = ↵ s exp ! (0) y is small

triple pomeron coupling pomeron intercept

10 BFKL pomeron loops

1 1 2 !(⌫,n)(Y Y 0) G(Y,r1,r2; Y 0,r10 ,r20 )= d⌫d r0e f(n, ⌫)E⌫⇤n(r100,r200)E⌫n(r10,r20) n= X1 Z1

e!(0)Y / Naive estimation shows: e!(0)Y (1 + ↵2e!(0)Y ) / s

2 = ↵ s exp ! (0) y plays a role of parameter in expansion in the number of loops For supercritical BFKL pomeron ! (0) > 0 . Parameter grows with rapidity

1) loop contribution becomes not small 2) one can not apply the perturbative approach

11 BFKL pomeron in the external ﬁeld of the nucleus Calculations of loops may become easier if one starts with the perturbative approach inside the nucleus from the start

@ = † + H + †( + †)+g⇢ L @y we make a shift in the quantum ﬁeld

†(y, b)=1†(y, b)+0†(y, b)

The model in the nuclear background ﬁeld:

2 = †(S +2 ) + + †( + †) L 1 0 0 1 1

M.A. Braun, A.N. Tarasov, Eur. Phys. J. C 58 (2008) 383

12 BFKL pomeron in the external ﬁeld of the nucleus

The nuclear ﬁeld transforms the supercritical pomeron into a subcritical one with the intercept smaller than unity.

For the BFKL pomeron we use effective non-local ﬁeld theory propagator in the external ﬁeld

2 @P (y, r1,r2) ↵¯ 2 r12 = d r3 2 2 P (y, r1,r3)+P (y, r2,r3) P (y, r1,r2) @y 2⇡ r13r23 Z ⇣ (y, r ,r )P (y, r ,r ) (y, r ,r )P (y, r ,r ) 1 3 2 3 2 3 1 3 with boundary condition ⌘

2 2 2 2 M.A. Braun, A.N. Tarasov, Nucl. Phys. P (y = y0,r1,r2; y0,r10 ,r20 )= 1 2 (r1 r10 ) (r2 r20 ) B 851 (2011) 533 r r

13 BFKL pomeron

2 2 2 2 P (y, r ,r )= d r0 d r0 P (y, r ,r ; y0,r0 ,r0 ) (r0 ,r0 ) 1 2 1 2 1 2 1 2 r1r2 1 2 Z With the chosen set of initial conditions, the convoluted BFKL pomeron propagator vanishes at large rapidity distances 2 b2/c c r e 2 P (y =0,r ,r )=1 e 1 12 1 2

initial condition for evolution of the convoluted propagator 7

6 c1 = 10; c2 = 2 φ=π/2

Calculation is possible p only numerically 3

0 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100

r1=r2 [1/(GeV/c)]

14 BFKL pomeron in (2+1) dimensional QCD

Propagator in the external ﬁeld:

2 @P (y, r1,r2) ↵¯ 2 r12 = d r3 2 2 P (y, r1,r3)+P (y, r2,r3) P (y, r1,r2) @y 2⇡ r13r23 Z ⇣ (y, r ,r )P (y, r ,r ) (y, r ,r )P (y, r ,r ) 1 3 2 3 2 3 1 3 ⌘

2↵ N ✓(rmax r )✓(r rmin) s c 2,1 3 3 2,1 the kernel in (2+1)-dimensional QCD

We expect to ﬁnd analytical solution in (2+1) dimensional QCD

J. Bartels, V.S. Fadin, L.N. Lipatov, Nucl. Phys. B 698 (2004) 255

M.A. Braun, A.N. Tarasov, Nucl. Phys. B 863 (2012) 495

15 BFKL pomeron in (2+1) dimensional QCD

In terms of S-matrix Sr r (y)=1 r r (y) 2 1 2 1 BK equation Pomeron propagator

r2 r2 @Sr2r1 @Pr2r1 = dr0 Sr2r0 Sr0r1 Sr2r1 = dr0 Sr2r0 Pr0r1 + Pr2r0 Sr0r1 Pr2r1 @y r1 @y r Z ⇣ ⌘ Z 1 ⇣ ⌘

r21y r21y r2r1 (y)=e Sr2r1 Qr2r1 (y)=e Pr2r1 (y)

@ (y) @Q(y) = 2(y) = Q(y), (y) @y @y n o

1 1 1 (y)= (0)[1 y (0)] Q(y, y0)= (y0) (y)Q(y0,y0) (y) (y0) T (y, y0)Q(y0y0)T (y, y0) ⌘

Solution was found in matrix notation

16 Initial condition We shall ﬁrst study the Green function which satisﬁes

Gr2r1 r0 r0 (y0,y0)=(r2 r20 )(r1 r10 ) | 2 1

Q(y, y0)=T (y, y0)Q(y0,y0)T (y, y0)

The solution factorizes y0r0 yr21 Gr r r r (y, y0)=e 21 Tr r (y, y0)Tr r (y, y0) 2 1| 20 10 2 20 10 1

The expression is different from zero r only in the interval: 1 r2

gr2r1 r0 r0 (y, y0) = 0 only if r1

r10 r20

17 Initial condition

The initial condition for the BFKL pomeron propagator:

2 2 gr2r1 r0 r0 (y0,y0)= 1 2 (r2 r20 )(r1 r10 ) | 2 1 r r In the one-dimensional space the initial condition can be presented as:

gr2r1 r0 r0 (y0,y0)=(r2 r20 )✓(r2 r20 )(r10 r1)✓(r10 r1) | 2 1

Q(y, y0)=T (y, y0)Q(y0,y0)T (y, y0)

The solution factorizes r1 r2 r21(y y0) gr2r1 r r (y, y0)=e Ur (y, y0)Ur (y, y0) | 20 10 220 101

r y0r0 U (y, y0)= dr0(r r0)T (y, y0)e r r0 Z0

g (y, y0) = 0 only if r

18 Numerical results In our calculations we have taken:

r Sr(0) = e

r r r1 rm r rm r r1 yr yr yr yr g (y, r r )=e dr U (y, 0) dr U (y, 0) g (y, r r0 )=e dr U (y, 0)e 1 dr U (y, 0)e 2 1 ⌘ 21 1 r1 2 r2 2 ⌘ 21 1 r1 2 r2 Z0 Z0 Z0 Z0

1 100

1e-05 0.01

0.0001

1e-06 1e-10

1e-08 1 2 g g 1e-10 1e-15 1e-12

1e-14

1e-20 1e-16

1e-18

1e-25 1e-20 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 r r

Numerical results for =0 . 1 and y =1, 3, 5, 7, 9

19 Numerical results

The inﬂuence of the ﬁeld and the role of value of

1000 < 1 corresponds 100 to vanishing Sr(y)

10 γ=2.0 x 30.6; vacuum For > 1 S-matrix 1

Integrated g tends to unity

0.1

γ=0.1, x 54.1 0.01

0.001 1 2 3 4 5 6 7 8 9 10 y 1) With < 1 the propagator the propagator in the nuclear field goes to zero much faster than in the vacuum

2) With > 1 we do not observe any influence of the field

20 BFKL pomeron loops in (2+1) dimensional QCD The pomeron self-mass:

⌃(y, r ,r y0,r0 ,r0 ) 2 1| 2 1

max r2,r1 max r20 ,r10 2 2 { } { } = (8⇡↵sNc) dr3 dr30 g(y, r2,r3 y0,r20 ,r30 )g(y, r3,r1 y0,r30 ,r10 ) min r ,r min r ,r | | Z { 2 1} Z { 20 10 } r3 It is trivial to see that this expression is zero due to the properties of the pomeron propagator r30 r30

Pomerons cannot form loops in the nuclear field.

21 Conclussions

1) We have studied numerically the BFKL pomeron propagator in the external field created by the solution of the BK equation in the nuclear matter

2) We have found that for more or less arbitrary set of initial conditions the convoluted propagator vanishes at large rapidities. This gives reasons to believe that the propagator itself vanishes at large rapidities in the nuclear background

3) In 2+1 dimensional QCD, BFKL pomerons in the nuclear field cannot form loops in the nuclear field. This implies that the BK equation gives the complete solution to the quantum field theory of interacting pomerons