QCD AMPLITUDES WITH THE GLUON EXCHANGE AT HIGH ENERGIES (AND GLUON - - PowerPoint PPT Presentation

Young Researchers Workshop ”Physics Challenges in the LHC Era”

QCD AMPLITUDES WITH THE GLUON EXCHANGE AT HIGH ENERGIES (AND GLUON REGGEIZATION PROOF)

contributor: Reznichenko Aleksei co-authors: Fadin Victor, Kozlov Mikhail Frascati, May 2009

• 1. TERMINOLOGY: REGGEIZATION
• Reggeization of any particle assumes

the signaturized amplitude to acquire the asymptotic behavior like sj(t), when ex- changing this particle in the t-channel in Regge limit (t ≪ s). Here j(t) ≡ 1+ω(t) is the Regge trajectory with ω(0) = 0.

• Signature in the channel tl for multi-

particle production means the (anti-) symmetrization with respect to the sub- stitution si,j ↔ −si,j, for i < l ≤ j.

A′ A B′ B . . . . . . . . .

sk,n+1 sk,n sk,l s0,ks1,ks2,k P0

q1, c1 q2, c2 qk, ck qk+1, ck+1

P1 P2 Pk Pl Pn Pn+1

qn+1, cn+1

Re Fig.1 The amplitude for the process 2 → n + 2.

• Hypothesis of the gluon reggeization claims that in Regge limit the real part of the

NLA-amplitude 2 → n + 2 with negative signature and with octet in all ti-channels has universal form: ReAA′B′+n

AB

= ¯ ΓR1

A′A

n

• i=1

eω(q2

i )(yi−1−yi)

q2

i⊥

γJi

RiRi+1

• eω(q2

n+1)(yn−yn+1)

q2

(n+1)⊥

ΓRn+1

B′B ,

(1) where yi = 1

2 ln( k+

i

k−

i ) — particle (Pi) rapidities, and γJi

RiRi+1, ΓR P ′P — known effective vertices.

• 2. THE PROOF IDEA: BOOTSTRAP RELATIONS AND CONDITIONS
• Bootstrap relations: There is infinite number of necessary and sufficient conditions for

compatibility of the Regge amplitude form (1) with unitarity: Re

• 1

−2πi n+1

• l=k+1

discsk,l −

k−1

• l=0

discsl,k

• AA′B′+n

AB

• = 1

2(ω(tk+1) − ω(tk))ReAA′B′+n

AB

(2)

• Bootstrap conditions: There is finite number of identities making all bootstrap rela-

tions fulfilled. These conditions restrict effective vertices and the gluon trajectory. The main goal is to prove them hold true. – Elastic conditions constrain the ker- nel ( ˆ K) and the effective vertex de- scribing the transition of the initial particle (B) to the final one (B′): | ¯ B′B = gΓRn+1

B′B |Rω(qB⊥),

ˆ K|Rω(q⊥) = ω(q2

⊥)|Rω(q⊥),

– Inelastic conditions appear in ele- mentary (2 → 3) inelastic amplitudes.

. . .

J1 Jj−1

γ

Jj−1 Rj−1Rj

JjRj|

Jj

A′ A

Jn

B . . .

• B′

|B′B ˆ Jn ˆ Jj+1

Jj+1

. . . . . . . . .

e ˆ

KYj+1

e ˆ

KYn+1

γJ1

R1R2

¯ ΓR1

A′A

Fig.2 sj,n+1-channel discontinuity calculation via unitarity relation

| ¯ JiRi+1 + ˆ Ji |Rω(q(i+1)⊥) g q2

(i+1)⊥ = |Rω(qi⊥) g γJi RiRi+1,

(3)

• 3. INELASTIC CONDITION: | ¯

JiRi+1 — NLO IMPACT-FACTOR

• Impact factor of the jet production is the first component of the inelastic bootstrap

condition, intrinsically it appears as logarithmically non-enhanced term in the disconti- nuity (in s0,1 or s1,2 channel) for the process 2 → 3.

k, a r1, c1 r2, c2 q1, i J(k′) G

(a)

=

k, a r2, c2 q1, i r1, c1 k1 k2 G

(b)

+

k, a r2, c2 q1, i r1, c1 k1 −k2 G

(c)

+

k, a r2, c2 q1, i G′(k′) r1, c1 G

(d)

+

k, a r2, c2 q1, i G′(k′) r1, c1 G

(e)

+

k, a G′(k′) q1, i ˆ Kr r, c r1, c1 r2, c2 G

(f)

Further we consider the most nontrivial NLO impact-factor: Jj = G(k) GR1|G1G2 = GR1|G1G2v.c. + GR1|G1G2loop. (4)

GR1|G1G2v.c. = δ(q1⊥ − k⊥ − r1⊥ − r2⊥) 2k+

• G′
• γG′(C)

R1G1 ΓG2(B) GG′

+ γG′(B)

R1G1 ΓG2(C) GG′

− γG′(B)

R1G2 ΓG1(C) GG′

− γG′(C)

R1G2 ΓG1(B) GG′

+ γG′(B)

R1G1 ΓG2(B) GG′ ×

× 1 2(ω(1)(q1) − ω(1)(r1)) ln

• k2

⊥

(q1 − r1)2

⊥

• − ω(1)(r2)

2 ln k2

⊥

r2

2⊥

• − γG′(B)

R1G2 ΓG1(B) GG′

• r1 ↔ r2
• .

GR1|G1G2loop = δ(q1⊥ − k⊥ − r1⊥ − r2⊥) 2k+

• f=2g,q¯

q

γ{f}

R1G1ΓG2 G{f} − γ{f} R1G2ΓG1 G{f}

• dφ∆

{f} − ∆GR1| ˆ

KB

r |G1G2B.

• 4. INELASTIC CONDITION: ONE GLUON PRODUCTION OPERATOR
• One gluon production operator is the second component of the inelastic bootstrap

condition, describing the transition reggeon-reggeon state to gluon-reggeon-reggeon.

q1 − r1 r1 q2 − r r G(k)

=

r1 r q2 − r q1 − r1 G(k)

(a)

+

r1 r q2 − r q1 − r1 G(k)

(b)

+

q1 − r1 r1 r q2 − r G(k)

(c)

+

r1 r q2 − r q1 − r1 G(k)

(e)

+

G(k) r1 q1 − r1 q2 − r r

(f)

G′

1G′ 2| ˆ

J ∆

i |G1G2 = δ(r1⊥ + r2⊥ − ki⊥ − r′ 1⊥ − r′ 2⊥)[γJi G1G′

1δ(r2⊥ − r′

2⊥)r 2 2⊥δG2G′

2+

+ γJi

G2G′

2δ(r1⊥ − r′

1⊥)r 2 1⊥δG1G′

1] +

• G

yi+∆

• yi−∆

dzG 2(2π)D−1(γ{JiG}

G1G′

1 γ

G2G′

2

G

+ γG

G1G′

1γ

G2G′

2

JiG ).

• 5. PREVIOUSLY OBTAINED RESULTS:
• By the direct calculation we (V.S., M.G., A.V.) demonstrated that the inelastic bootstrap

condition was fulfilled being projected onto the colour octet in the t-channel: f ac′

1c′ 2

• G′

1G′ 2| ˆ

Ji |Rω(q(i+1)⊥) g q2

(i+1)⊥ + G′ 1G′ 2| ¯

JiRi+1

• = f ac′

1c′ 2G′

1G′ 2|Rω(qi⊥) g γJi RiRi+1.

• We introduced the operator formulation of the bootstrap reggeon formalism: for bootstrap

relations and conditions. Further we conjectured that the following condition is valid for arbitrary color representation: ˆ Ji |Rω(q(i+1)⊥) g q2

(i+1)⊥ + | ¯

JiRi+1 = |Rω(qi⊥) g γJi

RiRi+1.

(5)

• In V.S. Fadin, R. Fiore, M.G. Kozlov, A.V. Reznichenko, Phys. Lett.B 639 (2006) using

the direct discontinuity calculation through the unitarity in terms of our components

− 4i(2π)D−2δ(q(j+1)⊥ − qi⊥ −

l=j

• l=i

kl⊥) discsi,jAS

2→n+2 = ¯

ΓR1

A′A

eω(q1)(y0−y1) q2

1⊥

i

• l=2

γJl−1

Rl−1Rl

eω(ql)(yl−1−yl) q2

l⊥

• ×

× JiRi| j−1

• l=i+1

e

ˆ K(yl−1−yl) ˆ

Jl

• e

ˆ K(yj−1−yj)| ¯

JjRj+1  

n

• l=j+1

eω(ql)(yl−1−yl) q2

l⊥

γJl

RlRl+1

  eω(qn+1)(yn−yn+1) q2

(n+1)⊥

ΓRn+1

B′B ,

(6)

and applying granted elastic and inelastic NLO bootstrap conditions we proved all boot- strap relations to be fulfilled. So the last millstone on the way of reggeization proof was the validity of (5).

• 6. DIFFERENT COLOUR REPRESENTATIONS IN T-CHANNEL:

This last unproved bootstrap condition (Ji is one gluon) can be present in projected form: G′

1G′ 2| ˆ

Ji |Rω(q(i+1)⊥) g q2

(i+1)⊥ + G′ 1G′ 2| ¯

JiRi+1 = G′

1G′ 2|Rω(qi⊥) g γJi RiRi+1,

First of all, it was proved for the octet (the most important) in the t-channel. There is no r.h.s. for any other t-channel representations R = 8, since from the explicit form of |Rω(qi⊥): P(R)c1,c2

c′

1,c′ 2

G′

1G′ 2|Rω(qi⊥) g γJi RiRi+1 = 0.

(7) From the explicit form of the effective vertices it is easy to see that there are only THREE nontrivial colour structures into the operator of the gluon production G′

1G′ 2| ˆ

Ji |Rω(q(i+1)⊥) and into the impact-factor G′

1G′ 2| ¯

JiRi+1. The optimal choice is the “trace-based”: Tr[T c2T aT c1T i], Tr[T aT c2T c1T i], Tr[T aT c1T c2T i] (8)

• The first colour structure is symmetric with respect to c1, c2 and can be reduced:

Tr[T c2T aT c1T i] = N2

c

2 P(0) + Nc+2 2 P(27) + 2−Nc 2 P(Nc>3) Corresponding coefficient had not

been calculated before. It is the last problem on the reggeization proof way.

• The coefficient at second colour structure Tr[T aT c2T c1T i] is very similar to the octet case,

and whereby is considered to be calculated.

• The last coefficient (at Tr[T aT c1T c2T i]) can be easily obtained from the previous one.

• 7. RESULTS AND PLANS:
• We formulated the gluon reggeization proof in operating form through the bootstrap

approach based on unitarity.

• We proved all bootstrap conditions (necessary for reggeization proof) to be correct when

projecting on octet colour representation. But it is not sufficient for the final proof!

• We proved all bootstrap conditions to be correct for second and third colour structure

and for all colour structures in fermionic sector of the inelastic bootstrap condition.

• We calculated in the dimensional regularization all components of the last coefficient at

the symmetric colour structure for the inelastic bootstrap condition.

• For this structure we demonstrated the cancellation of all singular terms (collinear regu-

larization, 1

ǫ2, and 1 ǫ ), rational, and logarithmic terms.

• We are planing to demonstrate the cancellation for dilogarithmic and double logarithmic

terms in the inelastic bootstrap condition in the nearest future. THANKS FOR YOUR ATTENTION!