Regge limit and the soft anomalous dimension Simon Caron-Huot - - PowerPoint PPT Presentation

Regge limit and the soft anomalous dimension

Simon Caron-Huot

(McGill University) Galileo Galilei Institute: Amplitudes in the LHC era, Oct. 31th 2018

Based on: 1701.05241,1711.04850 & in progress with: Einan Gardi, Joscha Reichel & Leonardo Vernazza

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(GeV) s 10

2

10

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tot

σ 40 60 80 100 120 140 160

accelerator pp p p accelerator ARGO-YBJ Akeno Fly’s Eye Auger TOTEM

constrained

[fig: Menon& Silva `13]

Nonperturbative forward physics: total pp cross-section: ~p Area

probability to decay in two partons parton cascade wee parton target cross section

• Fig. 3-a
• Why does it grow?
• Relativity + quantum mechanics


• A cloud of virtual particles (pions, rho’s,..)


builds around the proton!

3

[cartoon: Gottsman, Level&Maor]

p

• Deep inelastic scattering/saturation (HERA,heavy ions)


small x = Regge, large Q2⇒perturbative


• Mueller-Navelet: pp->X+2jets, forward&backward

(∆η 1, ∆φ)

perturbative phenomenology of forward scattering:

for review of these: see 1611.05079

5

theoretical motivations:

One of few limits where perturbation theory
 can be resumed Retain rich dynamics in 2D transverse plane:


• toy model for full amplitude
• nontrivial function spaces
• predicts amplitudes and other observables in

• verlapping limits

6

Here we’ll focus on A2→2, but in QCD at finite Nc.
 It depends on:

• energy:
• IR regulator: 1/𝜁 ⇔ log(-t/𝜈2)
• color: CA, T2

s, T2 t, . . .

L ≡ log |s/t| −iπ/2

The (multi-)Regge limit at higher points has been extensively studied, especially in planar N=4 SYM. It reveals an amazing integrable system (next talk?)

Nice variables

• 1. Coupling runs with transverse momenta, 


not CM energy

αs(−t)

⇒ use

• 2. Crossing symmetry relates large-s & large-u limits

L ≡ 1 2 ✓ log −s − i0 −t + log −u − i0 −t ◆ → log

• s

t

• − iπ

2

⇒ use symmetrical combination

8

M(±)(s, t) = 1

2

⇣ M(s, t) ± M(s t, t) ⌘

Crossing symmetry: Project onto signature eigenstates: These simple definitions remove all i\pi’s. The following have nice& real coefficients: M(−)(L, ↵s(−t), ✏), 1 i⇡ M(+)(L, ↵s(−t), ✏)

9

exponentiation


• f IR div

BFKL
 resummation soft limit of BFKL
 = Regge limit of 𝜟soft 2→2 kinematic limits soft Regge soft-
 Regge

L ∼ log s −t ∼ log −t µ2 1/✏

BFKL redux

[Balitsky,Fadin,Kuraev,Lipatov ’76-78]

A simple, and correct, approach to high-energy scattering:
 replace each fast parton by a null Wilson line

U(x⊥) ≡ Pei

R ∞

−∞ dx+Aa +(x+,0−,x⊥)T a

... cloud of 
 radiated
 partons The subtlety: projectiles contain more than one parton

• +
• (+perms.)

d dη

=

• ‘shock’ = Lorentz-contracted target
• 45o lines = fast projectile partons
• Each parton crossing the shock gets a Wilson line

z1 z2

Transverse distribution depends on energy resolution

d dη UU ∼ g2 Z d2z0K(z0, z1, z2) ⇥ U(z0)UU − UU ⇤

• Well established and tested
• Now well understood at NLL
• Partial NNLL results

13

[Balitsky ’95, Mueller,
 Kovchegov, JIMWLK*]

*Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov& Kovner

[Balitsky&Chirilli ’07&’13; Kovner,Lublinsky&Mulian ’13; SCH ’14] [SCH&Herranen ’16; Henn& Mistlberger ’17; SCH,Gardi&Vernazza ’17]

The Balitsky-JIMWLK equation

d dη ⌘ H = αs 2π2 Z d2zid2zj d2z0 z0i·z0j z2

0iz2 0j

⇣ T a

i,LT a j,L + T a i,RT a j,R U ab ad(z0)

• T a

i,LT b j,R + T a j,LT b i,R

⌘

Main feature: index contractions preserve
 two global symmetries: SU(N)past x SU(N)future Spontaneously broken to diagonal in vacuum:

[Kovner& Lublinsky ’05]

‘Goldstone boson’ W = Reggeized gluon

[SCH ’13]

U(x⊥) = eigT aW a(x⊥) h0|U(x⊥)|0i = 1

past future BFKL : expand in W’s and study linearized evolution

. . . . . . . . . Di Dj αg

Multi-Regge exchanges are suppressed by coupling + + LL = one-Reggeon NLL (two W’s) NNLL + … ‘Regge pole’ ‘Regge cut’

ANLL ∝ Z dνc(ν)sE(ν) ALL ∝ sαg(t) t M(−) M(+) M(−)

(one-W exchange)

16

d dη       (W)1 (W)2 (W)3 (W)4 · · ·       =       g2 g4 g6 · · · g2 g4 g4 g2 · · · g4 g2 · · ·       ·       (W)1 (W)2 (W)3 (W)4 · · ·      

Leading BFKL and
 BKP kernels

required by symmetry of d/dη

n→n+k transitions: from LO B-JIMWLK

• Matrix is symmetrical: projectile/target symmetry
• Growth/saturation: off-diagonal can’t be ignored
• (‘Reggeon field theory’ which resums all, still elusive)

eigW aT a ∼ 1 + igWT + . . .

Perturbative structure of the BFKL Hamiltonian

• At NNLL, something new happens: 1 and 3

Reggeon states mix

• @ 2-loops: violation of Regge pole factorization 

• @ 3-loops: first check of mixing matrix

[Del Duca, Falcioni, Magnea & Vernazza ’14]

H1→3 H3→1 H3→3

We don’t actually compute these diagrams:
 the LO B-JIMWLK Hamiltonian gives us simple 2d integrals all the work is to find the color factors that
 multiply them, starting from the Hamiltonian.

Hk→k+2 = ↵2

s

3⇡ Z [dzi][dz0] Kii;0 (Wi−W0)xW y

0 (Wi−W0)z Tr

⇥ F xF yF zF a⇤

• W a

i

(3.11) + ↵2

s

6⇡ Z [dzi][dzj][dz0] Kij;0 (F xF yF zF t)abh (Wi−W0)xW y

0 W z 0 (Wj−W0)t

− W x

i (Wi−W0)yW z 0 (Wj−W0)t − (Wi−W0)xW y 0 (Wj−W0)zW t j

i 2 W a

i W b j

.

ˆ M(−,3,1)

ij→ij = ⇡2⇣

R(3)

A T2 s−u[T2 t , T2 s−u] + R(3) B [T2 t , T2 s−u]T2 s−u + R(3) C (CA)3⌘

ˆ M(0)

ij→ij

result for 2loops NNLL, in any gauge theory: Poles are consistent with IR exponentiation!

✔

⇣ ⌘ ˆ M(−,2)

ij→ij =

 D(2)

i

+ D(2)

j

+ D(1)

i

D(1)

j

+ ⇡2R(2)⇣ (T2

s−u)2 1 12(CA)2⌘

ˆ M(0)

ij→ij,

= (rΓ)2 ✓ 1 8✏2 + 3 4✏⇣3 + 9 8✏2⇣4 + . . . ◆ (3.

R(2) =

Color operator precisely corrects factorization violation! at 3-loops NNLL, we computed coefficients


• f 3 color structures:

✔

R(3)

A

= 1 16 (rΓ)3(Ia−Ic) = (rΓ)3 ✓ 1 48✏3 + 37 24⇣3 + . . . ◆ ✓ ◆

20

In N=4, we could fix it from [Henn& Mistlberger ’16] Upshot:


• new 3loop prediction in QCD, up to one constant

• machine set up to compute NNLL M(-) to


4&higher loops, modulo same constant.

H(3)

1→1 =

= N2

c

 − ⇣2 144 1 ✏3 + 49⇣4 192 1 ✏ + 107 144⇣2⇣3 + ⇣5 4 + O(✏)

• + N0

c

 0 + O(✏)

• In N=4, that constant is known.

Removing the ‘hat’ requires the 3-loop gluon Regge trajectory: H1→1, which affects only the RC color structure.

1701.05241

. . .

Back to NLL: high-order Solution in the soft limit

21

two-Reggeon exchange

M(+)

[1711.04850] = leading order
 evolution

(∼ α`

sL`−1)

22

Both increase transcendental weight by 1 Each rung = the BFKL Hamiltonian H2→2

ˆ H = (2CA − T2

t ) ˆ

Hi + (CA − T2

t ) ˆ

Hm

‘integration’ part:

ˆ HmΨ(p, k) = 1 2✏  2 − ✓p2 k2 ◆✏ − ✓ p2 (p−k)2 ◆✏ Ψ(p, k)

‘multiplication’ part:

ˆ HiΨ(p, k) = Z d22✏k0 rΓ(2π)22✏ f(p, k, k0) [Ψ(p, k0) − Ψ(p, k)]

23

Cases where eigenfunctions are known: [Lipatov]

• Color singlet dipoles (x-space conformal symmetry)
• Color adjoint (p-space ‘dual’ conformal symmetry)

Unfortunately, for d≠4 / other color reps., eigenfunctions are not known ⇒ iterative solution Evolution equation:

Ψ(`) = ˆ HΨ(`−1), Ψ(0) = 1.

Exact solution in adjoint channel: Ψ = 1

ˆ M(+,4)

NLL = −i⇡(B0)4

3! Z [Dk] p2 k2(p − k)2 ⇢ (CA − T2

t )3 Ωmmm(p, k)

+ (2CA − T2

t )(CA − T2 t )2 Ωmim(p, k)

• T2

s−u M(tree)

= i⇡ (B0)4 4! ⇢ (CA − T2

t )3

✓ 1 (2✏)4 + 175⇣5 2 ✏ + O(✏2) ◆ (2.32) + CA(CA − T2

t )2

✓ − ⇣3 8✏ − 3 16⇣4 − 167⇣5 8 ✏ + O(✏2) ◆ T2

s−u M(tree).

Outermost rungs are always easy (multiplication) Note this has both leading& subleading IR divergences 4-loop = single nontrivial integral

[SCH ’13]

25

How to predict the IR divergences at higher-loops? Facts:

• 1. Wavefunction is finite as ✏ → 0

Ψ(`)(p, k)

• 2. Evolution closes in soft limit:

Z

k→0

Ψ(`)(p, k)

⇒poles can only appear from final integration

lim

k→0 ψ(`)(p, k) ∼ ˆ

H lim

k→0 ψ(`−1)(p, k)

. . .

IR divergences only occur when a full rail goes soft!

ki → 0 ˆ Hi ✓p2 k2 ◆n✏ = − 1 2✏ Bn(✏) B0(✏) ✓p2 k2 ◆(n+1)✏ ˆ Hm ✓p2 k2 ◆n✏ = 1 2✏ "✓p2 k2 ◆n✏ − ✓p2 k2 ◆(n+1)✏#

Gamma-functions ⇒Soft wave function = polynomial in

✓p2 k2 ◆✏

27

The soft wavefunction can be easily computed to all orders, and integrated to order O(✏0) get truckload of Gamma-functions:

ˆ M(+,`)

NLL

• s = i⇡

1 (2✏)` B`

0(✏)

`! (1 − ˆ B−1) (CA − T2

t )`−1 `

X

n=1

(−1)n+1 ✓` n ◆ ×

n−2

Y

m=0

 1 − ˆ Bm(✏)2CA − T2

t

CA − T2

t

• T2

s−u M(tree) + O(✏0),

However, is not random: WF has to be finite

✏ → 0

ˆ M(+,`)

NLL

• s = i⇡

1 (2✏)` B`

0(✏)

`! (1 − ˆ B−1) ✓ 1 − ˆ B−1(✏)2CA − T2

t

CA − T2

t

◆−1 × (CA − T2

t )`−1 T2 s−u M(tree) + O(✏0).

Whole thing reducible to a geometric series!

¯ M(+,1)

NLL = i⇡

 1 2✏ + O(✏0)

• T2

s−uM(tree),

¯ M(+,2)

NLL = i⇡(CA T2 t )

2!  1 (2✏)2 + O(✏0)

• T2

s−uM(tree),

¯ M(+,3)

NLL = i⇡(CA T2 t )2

3!  1 (2✏)3 + O(✏0)

• T2

s−uM(tree),

¯ M(+,4)

NLL = i⇡(CA T2 t )3

4!  1 (2✏)4 1 2✏ ⇣3CA 4(CA T2

t ) + O(✏0)

• T2

s−uM(tree),

¯ M(+,5)

NLL = i⇡(CA T2 t )4

5!  1 (2✏)5 1 (2✏)2 ⇣3CA 4(CA T2

t )

1 2✏ 3⇣4CA 16(CA T2

t ) + O(✏0)

• T2

s−uM(tree).

iteration of
 lower loops single poles =
 soft anomalous dimension

29

H = ZIRM ZIR = Pe−

R µ

dλ λ Γs(αs(λ))

Recall exponentiation of IR divergences: , Finite as 𝜁→0 Note that 𝜁→0 limit of H and 𝜟s contain all physically

• bservable part of S-matrix


(these suffice to compute inclusive cross-sections, when using suitable phase-space subtractions: cf Lorenzo’s talk)


H= IR&UV renormalized scattering =

[Weinzeirl]

Notice similarity when renormalizing UV&IR operators

ZUV = Pe

R ∞

µ dλ λ γ(αs(λ))

Oren(x) = ZUVObare(x),

Both exponentiate for same reason:
 disparate length scales factorize from each other ZIR = Pe−

R µ

dλ λ Γs(αs(λ))

,

H = ZIRMIR−bare

H

• IR

31

[Gardi&Magnea; Neubert&Becher ’09] [Almelid,Duhr&Gardi ’15]

Γs = X

i6=j

γK(αs(λ)) 4 log −sij λ2 T a

i T a j −

X

i

γJi(αs(λ)) + ∆

dipole ansatz departure,
 starts at 3-loops Can be expanded in Regge limit:

Γ (αs(λ)) = ΓLL (αs(λ), L) + ΓNLL (αs(λ), L) + ΓNNLL (αs(λ), L) + . . .

ΓLL (αs(λ)) = αs(λ) π γ(1)

K

2 L T2

t = αs(λ)

π L T2

t .

[Del Duca, Duhr, Gardi, Magnea& White ‘11]

At LL, gluon Reggeization fixes 𝜟s from gluon trajectory:

32

Z(+)

LL

⇣s t , µ, ↵s(µ) ⌘ = exp ⇢↵s ⇡ 1 2✏ LT2

t

• The LL Z-factor is a simple exponential:

≃ s

αsCA 2πϵ

NLL = perturbation around that

ˆ M(+)

NLL = exp

⇢ − ↵s(µ) ⇡ B0(✏) 2✏ LT2

t

 Z(−)

NLL

⇣s t , µ, ↵s(µ) ⌘ H(−)

LL ({pi}, µ, ↵s(µ))

+ Z(+)

LL

⇣s t , µ, ↵s(µ) ⌘ H(+)

NLL ({pi}, µ, ↵s(µ))

• ,

(4.

no
 poles ⇒ single-poles give 𝜟NLL, higher poles explicitly predicted

NLL = −

Z p d exp ⇢ 1 2✏ ↵s(p) ⇡ L(CA − T2

t )

 1 − ✓p2 2 ◆

✏

Γ()

NLL (↵s()) M(tree) + O(✏0).

(4.16)

33

Γ(−,`)

NLL =

i⇡ (` − 1)! 1 − CA CA − T2

t

R

• x(CA − T2

t )/2

• !−1
• x`−1

T2

s−u .

All-order result:

1 = Γ3(1 − ✏)Γ(1 + ✏) Γ(1 − 2✏) − 1 = −2⇣3 ✏3 − 3⇣4 ✏4 − 6⇣5✏5 −

• 10⇣6 − 2⇣2

3

• ✏6 + O(✏7)

R(✏)

34

Γ(−,1)

NLL = i⇡ T2 s−u

Γ(−,2)

NLL = 0

Γ(−,3)

NLL = 0,

Γ(−,4)

NLL = −i⇡ ⇣3

24 CA(CA − T2

t )2 T2 s−u,

Γ(−,5)

NLL = −i⇡ ⇣4

128 CA(CA − T2

t )3 T2 s−u,

Γ(−,6)

NLL = −i⇡ ⇣5

640 CA(CA − T2

t )4 T2 s−u,

Γ(−,7)

NLL = i⇡ 1

720 ⇣2

3

16 C2

A(CA − T2 t )4 + 1

32

• ⇣2

3 − 5⇣6

• CA(CA − T2

t )5

• T2

s−u,

Γ(−,8)

NLL = i⇡

1 5040 3⇣3⇣4 32 C2

A(CA − T2 t )5 + 3

64 (⇣3⇣4 − 3⇣7) CA(CA − T2

t )6

• T2

s−u.

…

• 1. only classical zeta’s, no zeta2.
• 2. Coefficients decay factorially

• 0.0

0.5 1.0 1.5 2.0

• 20
• 15
• 10
• 5

5 10 15 Γ(−)

NLL = i⇡↵s

⇡ G ⇣↵s ⇡ L ⌘ T2

s−u

ction for the expansion coefficients

is entire function

= αsL π

36

Asymptotics: G(x) ! c eax cos (bx + d)

• 5

10 15 20

• 0.2
• 0.1

0.0 0.1 0.2

• 5

10 15 20

• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6

Note: sign of 𝜟 itself is dominated by 𝜟LL Efficient evaluation via inverse Borel: G(x) = 1 2⇡i Z w+i∞

w−i∞

d⌘ g ✓1 ⌘ ◆ e⌘x

Finite part

37

Recall all physical info is in 𝜁→0 limit of H and 𝜟s

H = ZIRM

fully understood
 @ NLL ?? BFKL ladders

38

Claim: 𝜁→0 limit determined from evolution with 𝜁=0

H(+)

NLL =

• k soft

d2−2ϵkΨ(p, k) − (subtractions) +

• k hard

d2kΨ(p, k)

• ϵ=0

First line computable using soft limit


• f wavefunction in d dimensions

Second line: wavefunction =sum of SVHPLs

39

ˆ H = (2CA − T2

t ) ˆ

Hi + (CA − T2

t ) ˆ

Hm

‘integration’ & multiplication parts: In principle, we would like to diagonalize H:

ˆ Hi (z, ¯ z) = 1 4⇡ Z d2wK(w, ¯ w, z, ¯ z) [ (w, ¯ w) − (z, ¯ z)] ˆ Hm (z, ¯ z) = j(z, ¯ z) (z, ¯ z)

simple kernels:

j(z, ¯ z) = 1 2 log  z (1 − z)2 ¯ z (1 − ¯ z)2

• K(w, ¯

w, z, ¯ z) = 1 ¯ w(z − w) + 2 (z − w)(¯ z − ¯ w) + 1 w(¯ z − ¯ w).

40

• wf(1) → 1

2c2(L({0}) + 2L({1}))

• wf(2) → 1

4c1c2(−L({0, 1}) − L({1, 0}) − 2L({1, 1}))

+ 1

2c22(L({0, 0}) + 2L({0, 1}) + 2L({1, 0}) + 4L({1, 1}))

• It turns out we can ‘integrate-by-parts’ derivatives


without changing kernel z d dz h ˆ HiΨ(z, ¯ z) i

Z = ˆ Hi  z d dz Ψ(z, ¯ z)

• (full algorithm requires (1-z)d/dz, just a bit harder)

That way we easily generate SVHPL expressions

41

Mfinite(1) = 0 Mfinite(2) = 0 Mfinite(3) = 1 4(−11)c22ζ(3)

we can get the IR renormalized amplitude to very high

• rder

… At 11-loops, we do get SVZ5,3,3 Coefficient grow exponentially:
 finite radius of convergence in series seems alternating, for unitary representations Tt2>0

αsL

Conclusions

• Modern approach to high-energy scattering via

Wilson lines: new theoretical control @NNLL

• Systematic and now well-tested theory, simplifies and

exponentiate many diagrams in the forward limit


• Possible applications to

• Mueller Navelet jets, small-x physics

• Predictions and new techniques for fixed-order 


multi-loop QCD computations