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The Pomeron in QCD

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The Pomeron in QCD Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher The Pomeron in QCD Douglas Ross University of Southampton 2014 QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Topics to be Covered ◮ Introduction to Regge Theory- the “classical” Pomeron ◮ Building a Reggeon in Quantum Field Theory ◮ The “reggeized” gluon ◮ The QCD Pomeron - the BFKL Equation ◮ Some Applications ◮ Diffraction and the Colour Dipole Approach ◮ Running the Coupling ◮ Higher Order Corrections in BFKL ◮ Soft and hard Pomerons QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Strong Interactions Before QCD Extract information from unitarity and analyticity properties of S-matrix. Unitarity - Optical Theorem . ℑmAαα = ∑AαnAn∗α n 1 ℑmA(s,t = 0) = σ 2s TOT Can be extended (Cutkosky Rules) ∆s Aαβ = ∑ AαnAn∗β cuts Leads to self-consistency relations for scattering amplitudes (Bootstrap) QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Partial Wave Analysis b c J a d ab cd A → (s,t) = ∑aJ(s)PJ (1 2t/s), [cosθ = (1 2t/s), mi 0] J − − → Crossing: ac¯ bd¯ ab cd A → (s,t) = A → (t,s) = ∑aJ(t)P(J,1 2s/t) J − In the limit s t (diffractive scattering) ≫ P(J,1 2s/t) sJ − ∼ so that ac¯ bd¯ s ∞ J A → (s,t) → ∑bJ(t)s → J QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Sommerfeld-Watson Transformation iπJ ¯ (2J + 1) η + e Aac¯ bd(s,t) = aη(J,t)P(J,2s/t) → I ∑ C η= 1 sinπJ 2 ± plane l Poles at integer J C Deform contour to ′ For large n 0 C C s, integral along ′ is zero. Pick up only contributions C 1 from poles in J-plane. 2 N.B. Important (and possibly incorrect) assumption is that all singularities in J-plane are poles. η + eiπαi(t) s ∞ αi(t) A(s,t) → ∑ βi(t)s → i 2 αi are the poles in the J-plane For s ∞ we only need the leading pole. → QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Chew-Frautschi Plot 6 Continue to negative t ac a c 4 Reggeon 2 b d bd 0 0 2 4 6 8 αR(t) A γac(t)γbd(t)s ∼ (Factorisation) We can think of Regge exchange as the superposition of the exchange of many particles. QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher The Pomeron Leading trajectory Using Optical Theorem s ∞ α 0 A(s,0) → s ( ) →∼ implies α(0) 1 σTOT s − ∼ Okun-Pomeranchuk Theorem If exchanged Regge trajectory does NOT have the quantum numbers of the vacuum, α(0) < 1. Foldy-Peierls Theorem If α(0) 1, the Regge trajectory exchanged MUST have the quantum numbers≥ of the vacuum. THIS IS THE POMERON N.B. α(0) > 1 is NOT allowed by unitarity. Froissart-Martin bound (derived from unitarity) 2 σTOT < Aln s QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Landshoff-Donnachie fit 2 αP(t) = 1.08 + 0.25(GeV− )t QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Landshoff-Donnachie fit 2 αP(t) = 1.08 + 0.25(GeV− )t QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Landshoff-Donnachie fit 2 αP(t) = 1.08 + 0.25(GeV− )t QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Confronting Regge Theory with QCD ◮ How is the Pomeron explained in QCD? ◮ Data fitted by Landshoff-Donnachie cannot be calculated in purely perturbative QCD - non perturbative effects must be included ◮ Expect a hint of the Pomeron from perturbative QCD - but we don’t really find one ◮ Phenomenological models have to be used to explain data. ◮ Nevertheless perturbative QCD in the kinematic regime where Pomerons are expected to dominate give some interesting results (solutions to the BFKL equation) which can be compared with data in certain cases. QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Toy Model 3 LI = λφ Use Optical theorem to calculate ℑmA(√s,q) 0 l p p 1 1 @ @ Leading order k k + q @ 0 0 Neglecting masses. l p p 2 2 @ + k = (k ,k−,k) q = (0,0,q) 1 + 2 + 2 dLIPS = dk dk−d kδ(k (√s k−) k ) 2 − − + 2 + 2 4 + 2 δ(k (√s k−) k )δ(k−(√s k ) k ) ℑmA = (2π)λ dk dk−d k − − − − Z (k+k k2)(k+k (k q)2) − − − − − 1 d2k ≈ Z k2(k q)2 − QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Higher Orders In higher order we want to sum all terms λ2n lnn(s). Is this approximation valid? ∼ Maybe not - renormalon studies suggest that leading logarithm sums are not reliable. QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher 0 p p 1 1 @ @ k k + q 1 1 @ @ @ k k + q 2 2 @ 0 p p 2 2 @ 6 λ + 2 + 2 + 2 dk dk−d k1dk dk−d k2δ(k (√s k−) k1 ) 8π Z 1 1 2 2 1 − 1 − + 2 + 2 δ((k2 k1) (k2 k1) (k1 k2) )δ(k− (√s k ) k2 ) − − − − − 2 − − 2 − + 2 + 2 + 2 + 2 (k k− k1 )(k k− (k1 q )(k k− k2 )(k k− (k2 q) ) 1 1 − 1 1 − − 2 2 − 2 2 − − √s dk+ d2k d2k = 2 1 2 Z + + + 2 2 2 2 k (k k ) k1 (k1 q) k2 (k2 q) 1 2 − 1 − − + Integral over k2 gives ln(s) Dominated by the region √s k+ k+. ≫ 2 ≫ 1 QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher On the other hand . etc . give an extra power of λ2, but NO extra ln(s). Such graphs may be dropped in the leading log approximation. QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Multi-rung Ladder Graph @ @ @ @ @ √s/2 dk+ √s/2 dk+ @ n n 2 . ℑmA . + + + + @ Z + Z + ∼ kn 1 (kn kn 1) ··· k1 (k2 k2 ) @ k 1 i − − − − @ n @ ln s k i @ + @ Dominant region of k integration @ k i+1 @ + + + + . @ . √s k k k k . n n 1 2 1 @ ≫ ≫ − ···≫ ≫ @ “Multi-Regge regime” @ @ @ In this model only uncrossed ladder graphs contribute to the leading log approximation. QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Multi-rung Ladder Graph @ √s k+ k+ k+ k+ @ n n 1 2 1 ≫ ≫ − ···≫ ≫ @ @ Similarly @ @ . √s k− k− k− k− @ n n 1 2 1 ≪ ≪ − ···≫ ≫ @ k i1 @ The on-shell delta functions for cut lines @ k i @ + 2 @ δ(ki−+1ki (ki ki 1) ) @ − − − k i+1 @ . @ Propagators of the vertical lines . @ @ 1 1 + 2 @ + 2 2 (since ki ki /ki−) @ ki ki− ki ≈−ki ≪ @ − In this model only uncrossed ladder graphs contribute to the leading log approximation. QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Crossed-rung Graphs @ k + q k i1 i1 @ @ @ @ @ l k i @ @ @ @ k i+1 @ 2 2 + + + 2 l = (ki + ki+1 ki 1) = (ki + ki+1 ki 1)(ki− + ki−+1 ki− 1) ki − − − − − − −∼ + + 2 ki+1ki− 1 ki+1ki− k ≈ − ≫ ≫ In the multi-Regge region crossed-ladder graphs are suppressed because denominators of propagators are larger. QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Integral Equation . A = + A . 2 √s/2 + 2 λ dk d k + ℑmA(√s,q) = ℑmA0 + ℑmA(k ,q) 16π3 Z k+ k2(k q)2 − effect of↑ adding rung. QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Mellin Transform (Equivalent to Sommerfeld-Watson transformation) ω 1 ∞ s − − A˜ (ω) = A(s)ds Z s s0 0 If A(s) sω0 ∼ 1 A˜ (ω) ∼ (ω ω0) − Mellin transform has a pole at ω = ω0. Furthermore ω 1 ω 1 ∞ s − − ∞ s − − A˜ (ω)B˜ (ω) = A(s)ds ′ B(s )ds Z s Z s ′ ′ s0 0 s0 0 ω 1 + ∞ s − − dk √s = A(k+)B Z s Z k+ k+ s0 0 QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Integral Equation in Mellin Space λ2 1 d2k ℑmA˜ (ω) = ℑmA˜ 0 + ℑmA˜ (ω) 16π3 ω Z k2(k q)2 − Solution: 1 ℑmA˜ (ω) ∼ (ω ω0) − ℑmA(√s,q) sω0 ∼ λ2 d2k ω0 = 16π3 Z k2(k q)2 − QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher QCD ◮ Vertices contain momenta - cannot just consider uncrossed ladder graphs ◮ Need to account for tree graphs and loop corrections for cut graphs - “ladders within ladders” - bootstrap (self-consistency relations) ◮ Need to account for colour factors - distinguish between colour singlet and colour octet exchange QCD Pomeron University of Southampton Reggeons in QFT The reggeized Gluon The BFKL Equation Applications Colour Dipoles Running Coupling Higher Colour Octet Exchange 0 1 1 2 p 1 For q s use eikonal approximation | | ≪g .
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