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Integrability in the Multi- Regge Regime. Amplitudes 2013, Ringberg. Volker Schomerus DESY Hamburg. Based on work w. Jochen Bartels, Jan Kotanski , Martin Sprenger , Andrej Kormilitzin , 1009.3938, 1207.4204 & in preparation . Introduction . - PowerPoint PPT Presentation
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Integrability in the
Multi-Regge Regime
Volker SchomerusDESY Hamburg
Based on work w. Jochen Bartels, Jan Kotanski , Martin Sprenger,
Andrej Kormilitzin, 1009.3938, 1207.4204 & in preparation
Amplitudes 2013, Ringberg
Introduction
Goal: Interpolation of scattering amplitudes from weak to strong coupling
N=4 SYM: find remainder function R = R (u) cross ratios
From successful interpolation of anomalous dimensions
→ String theory in AdS can provide decisive input integrability at weak coupling not enough
Introduction: High Energy limit
Main Message: HE limit of remainder R at a=∞ is
determined by IR limit of 1D q-integrable system
Weak coupl: HE limit computable ← integrabilityBFKL,BKP
TBA integral eqs algebraic BA eqse.g.
Useful to consider kinematical limits: here HE limit [↔ Sever’s talk]
Main Result and Plan1. Multi-Regge kinematics and regions
2. Multi-Regge limit at weak coupling
(N)LLA and (BFKL) integrability, n=6,7,8…
3. Multi-Regge limit at strong coupling
• MRL as low temperature limit of TBA
• Mandelstam cuts & excited state TBA
• Formulas for MRL of Rn ,n=6,7 at a=∞
Cross ratios, MRL and regions
Kinematics
1.1 Kinematical invariants
t1
t2
t4 s4
s
s12
s123
2 → n – 2 = 5 production amplitude
t3 s3
s2
s1
½ (n2 -3n)
Mandelstam
invariants
1.1 Kinematical invariants
1.2 Kinematics: Cross Ratios
u3
1
u3
2u1
1 u1
2 u2
2
u2
1
u½ (n2 -5n)
basic cross
ratios (tiles) 3(n-5)
fundamental
cross ratios from
Gram det
1.3 Kinematics: Multi-Regge Limit
-ti << si xij ≈ si-1..sj-3
small
large
larger
1.4 Multi-Regge Regions2n-4 regions depending on the sign of ki0 = Ei
u2σ > 0 u3σ > 0 u2σ < 0 u3σ < 0
s1 < 0 s12 > 0 s123 < 0
s4 < 0 s34 > 0 s234 < 0
s1 > 0 s12 > 0 s123 > 0
s4 > 0 s34 > 0 s234 > 0
Weak Coupling
Weak Coupling: 6-gluon 2-loop
[Lipatov,Prygarin]
2-loop n=6 remainder function R(2)(u1,u2,u3) known [Del Duca et al.] [Goncharov et al.]
leading log
discontinuity
Continue cross ratios along
MHV
Leading log approximation LLA The (N)LLA for can be obtained from
Impact factor Φ & BFKL eigenvalue ω known in (N)LLA
Explicit formulas for R in (N)LLA derived to 14(9) loops[Dixon,Duhr,Pennington] all loop LLA proposal using SVHP [Pennington]
[Bartels, Lipatov,Sabio Vera]
[Fadin,Lipatov]
LLA: [Bartels et al.]
([Lipatov,Prygarin])
H2 and its multi-site extension ↔ BKP Hamiltonian
are integrable
LLA and integrability
[Faddeev, Korchemsky]
ω(ν,n) eigenvalues of `color octet’ BFKL Hamiltonian
BFKL Greens fct in s2 discontinuity
← wave fcts of 2 reggeized gluons
[Lipatov]
↔ integrability in color singlet case = XXX spin chain
H2 = h + h*
Beyond 6 gluons - LLAn=7: Four interesting regions
(N)LLA remainder involves the
same BFKL ω(ν,n) as for n = 6 [Bartels, Kormilitzin,Lipatov,Prygarin]
n=8: Eleven interesting regions
Including one that involves the
Eigenvalues of 3-site spin chain
?
paths
Strong Coupling
3.1 Strong Coupling: Y-System
Scattering amplitude → Area of minimal surface [Alday,Gaiotto, Maldacena][Alday,Maldacena,Sever,Vieira]
A=(a,s) a=1,2,3; s = 1, …, n-5 `particle densities’
rapidity
R = free energy of 1D quantum system involving 3n-15
particles [mA,CA] with integrable interaction [KAB ↔ SAB] complex masses chemical potentials
R = R(u) = R(m(u),C(u)) by inverting
R
Wall crossing & cluster algebras
3.2 TBA: Continution & Excitations [Dorey, Tateo]
Continue m along a curve in complex plane to m’ R
Solutions of = poles in integrand sign
contribution from excitations
Excitations created through change of parameters
3.3 TBA: Low Temperature LimitIn limit m → ∞ the integrals can be ignored:
Bethe Ansatz equations
energy of bare excitations
In low temperature limit, all energy is carried by
bare excitations whose rapidities θ satisfy BAEs.
= large volume L => large m = ML ; IR limit
,
3.4 The Multi-Regge Regime[Bartels, VS, Sprenger] Multi-Regge regime reached when
Casimir energy vanishes
at infinite volume
[Bartels,Kotanski, VS]n=6 gluons:
u1→ 1u2,u3 → 0
∞
while keeping Cs and fixed
4D MRL = 2D IR
using check
6-gluon casesystem parameters solutions of Y3(θ) = -1 as function of ϕ
6-gluon case (contd)
solutions of Y1(θ) = -1solutions of Y2(θ) = -1
Solution of BA equations with 4 roots θ(2) = 0, θ3 = ± i π/4
n > 6 - gluons
[Bartels,VS, Sprenger ]
in prep.
Same identities at in LLA at weak coupling
n=7 gluons:
n = 7 gluons (contd)
n > 6 - gluons
[Bartels,VS, Sprenger ]
in prep.
Same identities as in LLA at weak coupling
n=7 gluons:
is under investigation….
General algorithm exists to compute remainder fct.
for all regions & any number of gluons at ∞ coupling
involves same number e2 ?
Conclusions and Outlook
Multi-Regge limit is low temperature limit of TBA natural kinematical regime Simplifications: TBA Bethe Ansatz
Mandelstam cut contributions ↔ excit. energies
Regge regime is the only known kinematic limit in
which amplitudes simplify at weak and strong coupling
Regge Bethe Ansatz provides qualitative and quantitative
predictions for Regge-limit of amplitudes at strong coupling
Interpolation between weak and strong coupling ?
Two new entries in AdS/CFT dictionary: