Two- Gluon Correla.ons in Heavy- Light Ion Collisions

Transcription

1 Two- Gluon Correla.ons in Heavy- Light Ion Collisions Yuri V. Kovchegov The Ohio State University work in collabora.on with with Douglas Wertepny arxiv:22.95 [hep- ph], arxiv: [hep- ph]

2 Outline Long- range rapidity correla.ons in hadronic and nuclear collisions: general discussion Two- gluon produc.on in heavy- light ion collisions: Geometric correla.ons Long- range rapidity correla.ons Even harmonics Energy independence Ini.al vs final state correla.ons Perturba.ve HBT correla.ons k T - factoriza.on Conclusions

3 Introduc.on to satura.on physics Published in September 202 by Cambridge U Press

4 Long- range rapidity correla.ons

5 Ridge in heavy ion collisions Heavy ion collisions, along with high- mul.plicity p+p and p+a collisions, are known to have long- range rapidity correla.ons, known as the ridge : Au+Au central 3<p t trig <4 GeV/c N/(dΔφ dδη) 2 d Δφ Δη 0.5.5

6 Origin of rapidity correla.ons =const k t k 2 x =const x + Causality demands that long-range rapidity correlations originate at very early times (cf. explanation of the CMB homogeneity in the Universe) x 3 A A 2 Gavin, McLerran, Moschelli 08; Dumitru, Gelis, McLerran, Venugopalan 08.

7 Correla.ons in AdS shock wave collisions µ µ k k k 2 k 2 µ 2 µ 2 I II C(k,k 2 ) cosh (4 y) H. Grigoryan, Yu.K. 0

8 Correla.ons in AdS shock wave collisions C(k,k 2 ) cosh (4 y) Correla.ons grow with rapidity interval??? It is possible that higher- order correc.ons in shock wave strengths will modify this result, making it closer to real life. However, such correc.ons are important at later.mes, and are less likely to affect the long- range rapidity correla.on This could be another argument in favor of weakly- coupled dynamics in the early stages of heavy ion collisions.

9 Ridge in CGC There are two explana.ons of the ridge in CGC: Long- range rapidity- independent fields are created at early.mes, with correla.ons generated soon ader and with azimuthal collima.on produced by radial hydro flow. (Gavin, McLerran, Moschelli 08) Both long- range rapidity correla.ons and the azimuthal correla.ons are created in the collision due to a par.cular class of diagrams referred to as the Glasma graphs.

10 Glasma graphs Generate back- to- back and near- side azimuthal correla.ons. Dumitru, Gelis, McLerran, Venugopalan 08.

11 Glasma graphs in LC gauge Glasma graphs are one of the many rescalering diagrams when two nucleons with a gluon each scaler on a nuclear target.

12 What to calculate? To systema.cally include Glasma graphs in the CGC formalism it would be great to solve the two- gluon inclusive produc.on problem in the MV model, that is, including mul.ple rescalerings in both nuclei to all orders (the two produced gluons only talk to each other through sources): A very hard problem!

13 Heavy- Light Ion Collisions Lille steps for the lille feet: consider mul.ple rescalerings only in one of the two nuclei.

14 Double gluon produc.on in heavy- light ion collisions

15 A. Senng up the problem: geometric correla.ons

16 Two- gluon produc.on We want to calculate two gluon produc.on in A +A 2 collisions with << A << A 2 resumming all powers of s 2 A /3 2 while (mul.ple rescalerings in the target nucleus) s 2 A /3 projec.le A target A 2 The gluons come from different nucleons in the projec.le nucleus as A >> and this is enhanced compared to emission from the same nucleon.

17 Applicability region The satura.on scales of the two nuclei are very different: QCD Q s Q s2 We are working above the satura.on scale of the smaller nucleus: k T Q s We thus sum all mul.ple rescalerings in the larger nucleus, Q s2 /k T ~, staying at the lowest non- trivial order in Q s /k T <<. Mul.ple interac.ons with the same nucleon in either nucleus are suppressed by Λ QCD /k T <<<.

18 Produc.on cross- sec.on The single- and double- inclusive cross sec.ons can be wrilen as Z d d d 2 kdy = d 2 Bd 2 b I(B b) 2 pa 2 d 2 kdyd 2 b d d 2 k dy d 2 k 2 dy 2 = Z d 2 Bd 2 b d 2 b 2 II(B b, B b 2 ) 2 d pa 2 d 2 k dy d 2 b d pa2 d 2 k 2 dy 2 d 2 b 2 Assume a large nucleus with uncorrelated nucleons (MV/Glauber model). Then the single- and double- nucleon wave func.ons are (with T the nuclear profile func.on) b I(b) 2 = T (b) II(b, b 2 ) 2 = T (b ) T (b 2 ) B b 2 A A 2

19 Geometric Correla.ons d pa 2 d 2 k dy d 2 b d pa2 d 2 k 2 dy 2 d 2 b 2 d pa 2 d 2 k dy d 2 b d pa 2 d 2 k 2 dy 2 d 2 b 2 Assume uncorrelated interac.on with the target: For cross sec.ons we have Z d d 2 kdy = d 2 Bd 2 bt (B b) d d 2 k dy d 2 k 2 dy 2 = Clearly Z d 2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) d d 2 k dy d 2 k 2 dy 2 d d pa 2 d 2 k dy d 2 kdyd 2 b d pa 2 d 2 k dy d 2 b d d 2 k 2 dy 2 d pa 2 d 2 k 2 dy 2 d 2 b 2 Correla.ons due to the integra.on over the impact parameter B - > Geometric correla.ons!

20 Geometric correla.ons: physical meaning In the same even the two nucleons are always within the smaller nucleus diameter from each other (in transverse plane). In different events they can be anywhere in the larger nucleus. b B C / d d 2 k dy d 2 k 2 dy 2 d d d 2 k dy d 2 k 2 dy 2 A 2 b 2 A

21 Geometric correla.ons at fixed B are zero as d d 2 k dy d 2 k 2 dy 2 d 2 B = d d 2 k dy d 2 B d d 2 k 2 dy 2 d 2 B which is clear from d d 2 k dy d 2 k 2 dy 2 = Z d d 2 kdy = Z d 2 Bd 2 bt (B b) d 2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) d pa 2 d 2 kdyd 2 b d pa 2 d 2 k dy d 2 b d pa 2 d 2 k 2 dy 2 d 2 b 2 Note that direc.on of B has to be fixed (to remove the geometric correla.ons), in other words the vector B should be fixed with respect to vectors k and k 2. Maybe hard to do, but perhaps possible.

22 B. Two- gluon produc.on

23 (i) Single gluon produc.on in pa

24 Single gluon produc.on in pa Model the proton by a single quark (can be easily improved upon). The diagrams are shown below (Yu.K., A. Mueller 97): b k x y Mul.ple rescalerings are denoted by a single dashed line: Σ n =0 2 n x x =0

25 Single gluon produc.on in pa b k x y The gluon produc.on cross sec.on can be readily wrilen as (U = Wilson line in adjoint representa.on, represents gluon interac.ons with the target) d pa 2 d 2 kdyd 2 = Z s C F b 4 4 d 2 xd 2 ye i k (x y) x b x b 2 y b y b 2 Nc 2 Tr[U xu y] Nc 2 Tr[U xu b ] Nc 2 Tr[U bu y] +

26 Forward dipole amplitude The eikonal quark propagator is given by the Wilson line 2 3 Z V (x) =Pexp4ig dx + A (x +,x =0,x) 5 The eikonal quark propagator is given by the Wilson line x ± = t p ± z 2 N(x,x )= x x 2 N tr V (x ) V (x ) x

27 (ii) Two- gluon produc.on in heavy- light ion collisions

28 The process b x,λ,a x 2,λ,b b 2 A B C D E F Solid horizontal lines = quarks in the incoming nucleons. Dashed ver.cal line = interac.on with the target. Doled ver.cal lines = energy denominators (ignore).

29 Amplitude squared b k x y x 2 y 2 k 2 b 2 This contribu.on to two- gluon produc.on looks like one- gluon produc.on squared, with the target averaging applied to both.

30 Amplitude squared b x y 2 b 2 x 2 y b x y 2 These contribu.ons to two- gluon produc.on contain cross- talk between the emissions from different nucleons. y x 2 b 2

31 Two- gluon produc.on cross sec.on Squaring the single gluon produc.on cross sec.on yields d d 2 k dy d 2 = 2 s CF 2 k 2 dy Z d 2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) d 2 x d 2 y d 2 x 2 d 2 y 2 e i k (x y ) i k 2 (x 2 y 2 ) x b x b 2 Nc 2 Tr[U x U y ] Nc 2 Tr[U x 2 U y 2 ] y b y b 2 x 2 b 2 x 2 b 2 2 y 2 b 2 y 2 b 2 2 Nc 2 Tr[U x U b ] N 2 c Tr[U x 2 U b 2 ] Nc 2 Tr[U b U y ] + Nc 2 Tr[U b 2 U y 2 ] + b k x y (cf. Kovner & Lublinsky, 2) x 2 y 2 k 2 b 2

32 Two- gluon produc.on cross sec.on The crossed diagrams give Z d crossed d 2 k dy d 2 = k 2 dy 2 [2(2 ) 3 ] 2 d 2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) d 2 x d 2 y d 2 x 2 d 2 y 2 he i i k (x y 2 ) i k 2 (x 2 y ) + e i k (x y 2 )+i k 2 (x 2 y ) 6 2 s 2 C F 2N c x b x b 2 y 2 b 2 y 2 b 2 2 x 2 b 2 x 2 b 2 2 y b y b 2 apple Q(x, y, x 2, y 2 ) Q(x, y, x 2, b 2 ) Q(x, y, b 2, y 2 )+S G (x, y ) Q(x, b, x 2, y 2 ) + Q(x, b, x 2, b 2 )+Q(x, b, b 2, y 2 ) S G (x, b ) Q(b, y, x 2, y 2 )+Q(b, y, x 2, b 2 ) + Q(b, y, b 2, y 2 ) S G (b, y )+S G (x 2, y 2 ) S G (x 2, b 2 ) S G (b 2, y 2 )+ b b x y2 x y2 x2 y y x2 b2 b2

33 Two- gluon produc.on cross sec.on The crossed diagrams give Z d crossed d 2 k dy d 2 = k 2 dy 2 [2(2 ) 3 ] 2 d 2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) d 2 x d 2 y d 2 x 2 d 2 y 2 he i i k (x y 2 ) i k 2 (x 2 y ) + e i k (x y 2 )+i k 2 (x 2 y ) 6 2 s 2 C F 2N c x b x b 2 y 2 b 2 y 2 b 2 2 x 2 b 2 x 2 b 2 2 y b y b 2 apple Q(x, y, x 2, y 2 ) Q(x, y, x 2, b 2 ) Q(x, y, b 2, y 2 )+S G (x, y ) Q(x, b, x 2, y 2 ) + Q(x, b, x 2, b 2 )+Q(x, b, b 2, y 2 ) S G (x, b ) Q(b, y, x 2, y 2 )+Q(b, y, x 2, b 2 ) + Q(b, y, b 2, y 2 ) S G (b, y )+S G (x 2, y 2 ) S G (x 2, b 2 ) S G (b 2, y 2 )+ We introduced the adjoint color- dipole and color quadrupole amplitudes: S G (x, x 2,y) Q(x, x 2, x 3, x 4 ) N 2 c N 2 c Tr[Ux U x 2 ] Tr[Ux U x 2 U x3 U x 4 ]

34 Two- gluon produc.on: proper.es d d 2 k dy d 2 = 2 s CF 2 k 2 dy Z d 2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) d 2 x d 2 y d 2 x 2 d 2 y 2 e i k (x y ) i k 2 (x 2 y 2 ) x b x b 2 Nc 2 Tr[U x U y ] Nc 2 Tr[U x 2 U y 2 ] y b y b 2 x 2 b 2 x 2 b 2 2 y 2 b 2 y 2 b 2 2 Nc 2 Tr[U x U b ] N 2 c Tr[U x 2 U b 2 ] Nc 2 Tr[U b U y ] + Nc 2 Tr[U b 2 U y 2 ] + Note that if the interac.on with the target factorizes, Tr[Ux U y ] Tr[U x2 U y 2 ] Tr[U x U y ] Tr[U x2 U y 2 ] large Nc, large A2 we s.ll have the geometric correla0ons. The geometric correla.ons lead to non- zero cumulants! (Like everything that depends on geometry.)

35 Two- gluon produc.on: proper.es d d 2 k dy d 2 = 2 s CF 2 k 2 dy Z d 2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) d 2 x d 2 y d 2 x 2 d 2 y 2 e i k (x y ) i k 2 (x 2 y 2 ) x b x b 2 Nc 2 Tr[U x U y ] Nc 2 Tr[U x 2 U y 2 ] y b y b 2 x 2 b 2 x 2 b 2 2 y 2 b 2 y 2 b 2 2 Nc 2 Tr[U x U b ] N 2 c Tr[U x 2 U b 2 ] Nc 2 Tr[U b U y ] + Nc 2 Tr[U b 2 U y 2 ] + If we expand the interac.on with the target to the lowest non- trivial order, one reproduced the contribu.on of the glasma graphs: d corr d 2 k dy d 2 k 2 dy 2 LO = 2 s 4 4 Z d 2 Bd 2 b [T (B away- side correla.ons b)] 2 Q4 s0(b) k 2 k2 2 Z d 2 l (l 2 ) 2 (k + k 2 ) 2 (cf. Dumitru, Gelis, McLerran, Venugopalan 08) apple (k l) 2 (k 2 + l) 2 + (k l) 2 (k 2 l) 2 near- side correla.ons (k k 2 ) 2

36 Two- gluon produc.on: proper.es b Crossed diagrams at lowest nontrivial order b2 x x2 y2 y k 2 + l k l l k k 2 + l k 2 k + l k l cos 2 apple (k k 2 ) 2 + (k + k 2 ) 2 l k 2 k 2 k + l Stronger correla.ons!

37 Two- gluon produc.on: proper.es d d 2 k dy d 2 = 2 s CF 2 k 2 dy Z d 2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) d 2 x d 2 y d 2 x 2 d 2 y 2 e i k (x y ) i k 2 (x 2 y 2 ) x b x b 2 Nc 2 Tr[U x U y ] Nc 2 Tr[U x 2 U y 2 ] y b y b 2 x 2 b 2 x 2 b 2 2 y 2 b 2 y 2 b 2 2 Nc 2 Tr[U x U b ] N 2 c Tr[U x 2 U b 2 ] Nc 2 Tr[U b U y ] + Nc 2 Tr[U b 2 U y 2 ] + The cross sec.on is symmetric under (dilo for the crossed term) k $ k 2 k 2! k 2 (just coordinate relabeling) as Tr U x U y = Tr Uy U x Hence the correla.ons generate only even azimuthal harmonics cos 2 n ( 2 )

38 Correla.on func.on May look like this (a toy model; two par.cles far separated in rapidity, jets subtracted, pa and AA): C(Δφ) C Φ Δφ Δφ Φ Dumitru, Gelis, McLerran, Venugopalan 08; Kovner, Lublinsky 0; Yu.K., D. Wertepny 2; Lappi, Srednyak, and Venugopalan 09

39 LHC p+pb data from ALICE These are high- mul.plicity collisions: it is possible that quark- gluon plasma is created in those collisions, with the hydrodynamics contribu.ng to these correla.ons. Satura.on approach is lacking the odd harmonics, like cos (3 Δφ), etc. Can they be generated by correc.ons to the leading- order CGC calcula.on?

40 Ini.al vs final- state correla.ons: U+U Hydro conven.onal wisdom: eccentricity is higher in side- by- side collisions, and the flow harmonics are larger. Satura.on approach: local satura.on scale is larger in.p- on-.p collisions, making correla.ons stronger as well. C tip on tip (k,y, k 2,y 2 ) LO C side on side (k,y, k 2,y 2 ) LO =.26 (for U + U) Qualita.vely different behavior in the two approaches! (~r) = 0 e x2 y 2 2 R 2 R 2 R 2 z2

41 Energy (In)dependence Glasma graphs correla.on func.on has the same number of Q s factors in the numerator and the denominator: C(k,y, k 2,y 2 ) LO / R d 2 Bd 2 b [T (B b)] 2 Q R 4 s2(b) d2 Bd 2 b d 2 b 2 T (B b ) T (B b 2 ) Q 2 s2 (b ) Q 2 s2 (b 2) Loosely- speaking, since each satura.on scale is propor.onal to a power of energy, Q 2 s s the energy dependence cancels, and the correla.on func.on is (almost) energy- independent!

42 C. HBT correla.ons

43 HBT diagrams There is another contribu.on coming from the crossed diagrams b x y 2 b 2 x 2 y

44 HBT diagrams They give HBT correla.ons (with R long =0 due to Lorentz contrac.on) k + l k l k + l l k k l 2 (k k 2 ) k 2 = k l l l Just like the standard HBT correla.ons (k ) 2(k 2 )+ (k 2 ) 2(k ) 2! (k ) 2(k 2 ) (k 2 ) 2(k )+c.c Possibly fragmenta.on would break phase coherence making these perturba.ve HBT correla.ons not observable.

45 Back- to- back HBT? Note that all our formulas are symmetric under k 2! k 2 Therefore, the HBT correla.on is accompanied by the iden.cal back- to- back HBT correla.on 2 (k + k 2 ) Note again that this correla.on may be destroyed in hadroniza.on.

46 k T - Factoriza.on?

47 In search of factoriza.on Since a lot of CGC ridge phenomenology was done using a k T - factorized expression, it is natural to try to see whether our two- gluon produc.on cross sec.on can be wrilen in a factorized form. The answer is yes, but factoriza.on does not come naturally.

48 Two- gluon Distribu.on Func.ons Unintegrated single- gluon distribu.on (gluon TMD) can be defined through a (gluon) dipole operator N G. d A (q,y) d 2 b A = C F s (2 ) 3 The two- gluon ditribu.ons can be defined either through a double- trace or the quadrupole operators: Z d 2 re iq r r 2 r N G (b + r, b,y)

49 Two- gluon Distribu.on Func.ons The two- gluon distribu.ons are: * + d D,Q A 2 (q, q 2,y) d 2 b d 2 b 2 A 2 = CF s (2 ) 3 2 Z d 2 r d 2 r 2 e iq r iq 2 r 2 r 2 r r 2 r 2 N D,Q (b + r, b, b 2 + r 2, b 2,y) where the double- trace and quadrupole operators are N D (x, y, z, w,y)= Tr Ux (Nc 2 ) 2 U y Tr Uz U w (Y ) A 2 N Q (x, y, z, w,y)= N 2 c Tr Ux U y U z U w A 2 (Y )

50 k T - Factoriza.on Ader some algebra the two- gluon inclusive produc.on cross sec.on can be wrilen in a factorized form: 2 Z d 2 s d 2 k dy d 2 = k 2 dy 2 C F k 2 k2 2 (* + d D A2 (q k, q 2 k 2,y) d 2 b d 2 b 2 A 2 + Z d d 2 Bd 2 b d 2 b 2 d 2 q d 2 A (q,y = 0) d A (q 2,y = 0) q 2 d 2 (B b ) A d 2 (B b 2 ) A " * + # K(b, b 2, k, k 2, q, q 2 ) d Q A 2 (q k, q 2 k 2,y) Nc 2 d 2 b d 2 +(k 2! k 2 ) b 2 A 2 with a somewhat involved (but known) coefficient func.on K. The expression is rather convoluted, and the func.on K depends on the nucleons posi.ons The expression is different from what was used in phenomenology.

51 Conclusions We completed a calcula.on of the two- gluon inclusive produc.on cross sec.on in nuclear collisions including satura.on effects in one nucleus to all orders (heavy- light ion collisions). Correla.ons we see: Geometric correla.ons HBT correla.ons (along with b2b HBT ones) Away- side correla.ons Near- side long- range rapidity correla.ons ( = long- range in rapidity) Near- and away- side correla.ons are iden.cal to all orders in satura.on effects: this is mainly consistent with the p+pb data from LHC. Odd harmonics are missing. Long- range rapidity correla.ons are (almost) energy- independent. k T - factoriza.on can be obtained with 2 types of double- gluon distribu.ons. It seems like geometric correla.ons are hard to remove by cumulants, since they depend on non- local geometry.

52 Backup Slides

53 Ridge in heavy ion collisions Heavy ion collisions, along with high- mul.plicity p+p and p+a collisions, are known to have long- range rapidity correla.ons, known as the ridge :

54 This conclusion is consistent with the data

55 This conclusion is consistent with the data