Production of a gluon with the exchange of three reggeized gluons in the Lipatov effective action approach

Abstract

In the Regge kinematics the amplitude for gluon production off three scattering centers is found in the Lipatov effective action technique. The vertex for gluon emission with the reggeon splitting in three reggeons is calculated and its transversality is demonstrated. It is shown that in the sum of all contributions terms containing principal value singularities are canceled and substituted by the standard Feynman poles. These results may be used for calculation of the inclusive cross section for gluon production on two nucleons in the nucleus.

Appendices

Appendix A: Calculation of the induced vertex

We denote the term

(A.1)

as

$$-\frac{[b1234]}{1(3+4)4} $$

for brevity.

We have to calculate the convolution of the induced vertex with the polarization vector of the gluon. Then the common factor

$$ -ig^3 \bigl(q^2_{\perp}\bigr) \epsilon^{*}_{-}= ig^3 \frac{2(p\epsilon^{*})_{\perp}}{p_{+}} q^2_{\perp} $$

(A.2)

includes (−1) from (A.1), i for the vertex, −g ³ from (22), (−i)⁵ from fields, $-q^{2}_{\perp}$ from $\partial^{2}_{\perp}$, (−i)³ from three derivatives $\partial^{-1}_{-}$ and the component $\epsilon^{*}_{-}(p)=-2(p\epsilon^{*})_{\perp}/p_{+}$ of the polarization vector is correspondent to V ₋.

Suppressing the factor (A.2) and collecting similar terms, for the vertex we get

(A.3)

Further, using these identities:

we get

(A.4)

We can rewrite (A.4) in the form of the expression containing the factor p ₋=q ₄₋ in the denominator of each term with the use of the identities

which are consequences of the relation q ₁₋+q ₂₋+q ₃₋+q ₄₋=0. Besides, we note that the T-matrices in the numerators form commutators [T ^a,T ^b]=if ^abc T ^c. This gives

(A.5)

for the vertex, where, for example, $f^{12c}\equiv f^{b_{1}b_{2}c}$.

Arranging the T-matrices in the commutators once and once more and substituting $\mathop {\mathrm {tr}}([T^{a},T^{b}]T^{c} )=\frac{i}{2}f^{abc}$, we get

(A.6)

Restoring the original notation and taking into account the factor (A.2), we finally get the answer

(A.7)

On the mass shell of the induced gluon the factor $p_{+}p_{-}=-p^{2}_{\perp}$ appears in the denominator, therefore the vertex remains finite in the limit p ₋→0.

The vertex can be expressed in the more convenient form. With the use of the Jacobi identity $f^{bcd}f^{b_{1}b_{2}c}=f^{bb_{2}c}f^{b_{1}cd}-f^{bb_{1}c}f^{b_{2}cd}$ and the relation

$$\frac{1}{q_{1-}q_{2-}}= \frac{1}{(q_{1-}+q_{2-})q_{1-}}+\frac{1}{(q_{1-}+q_{2-})q_{2-}} $$

the sum of the first and the fourth term in (A.7) can be rewritten as the sum

$$\frac{f^{bb_1c}f^{b_2cd}f^{ab_3d}}{(q_{1-}+q_{2-})q_{1-}} +\frac{f^{bb_2c}f^{b_1cd}f^{ab_3d}}{(q_{1-}+q_{2-})q_{2-}}. $$

With an analogous transformation of the rest terms, the vertex takes the form

(A.8)

The expression (A.8) is obviously symmetrical with respect to all permutations of momenta q _1,2,3 and correspondent indices b _1,2,3 of the induced reggeons.

Appendix B: Calculation of the four-reggeon vertex

Since the quadruple term  tr([v ₊,v ₋][v ₋,v ₋]) is absent in the Yang–Mills Lagrangian, the four-reggeon vertex 〈A ₊ A ₋ A ₋ A ₋〉 arises only from the term

(B.1)

of the induced part of the Lagrangian density (1).

To write the vertex in the momentum representation

(B.2)

we took into account the following factors: i for the vertex, g ² from (B.1), (−i)⁴ from fields, $-(q_{1} +q_{2} +q_{3})^{2}_{\perp}$ from $\partial^{2}_{\perp}$ and (−i)² from two derivatives $\partial^{-1}_{-}$.

Suppressing the common factor and denoting the typical term as

$$\frac{\mathop {\mathrm {tr}}(T^{c}T^{b_3}T^{b_2}T^{b_1})}{q_{1-}(q_{1-}+q_{2-})} \equiv \frac{[c321]}{1(1+2)} $$

we rewrite (B.2) in the form

(B.3)

Using the relation q ₁₋+q ₂₋+q ₃₋=0 and the identity

$$\frac{1}{q_{1-}q_{2-}}= \frac{1}{q_{1-}(q_{1-}+q_{2-})}+\frac{1}{q_{2-}(q_{1-}+q_{2-})} $$

we get

(B.4)

In the original notation, restoring the common factor, we get finally for the vertex

(B.5)

The expression for the diagram from Fig. 6 is given by the four-reggeon vertex, multiplied by the Lipatov vertex, two reggeon propagators and the quark-reggeon vertex and equals

(B.6)

what coincides with (41).

Appendix C: Proof of transversality of the R→RRRP vertex

3.1 C.1 Effective vertex in an arbitrary gauge

To verify the transversality of R→RRRP vertex it is necessary to write for it the explicit expression in an arbitrary gauge. As was shown such an expression consists of several parts. The part corresponding to Fig. 4(1) is

(C.1)

The part corresponding to Figs. 4(2) and 4(3) is

(C.2)

The part corresponding to Fig. 4(4) is

(C.3)

The part corresponding to Fig. 4(5) is

$$ -ig^3q_{\bot}^2n_{\mu}^- \frac{i}{2p_-} \frac{f^{b_3 ad}f^{b_2dc}f^{bb_1c}}{q_{1-}(q_{2-}+q_{1-})}. $$

(C.4)

So the total expression for V ₅-effective vertex in an arbitrary gauge is equal to the sum of expressions (C.1)–(C.4).

3.2 C.2 Proof of transversality

To prove the transversality of effective V ₅-vertex one should convolute an expression for it in arbitrary gauge with momentum vector p _μ . We will do it separately for each of the parts (C.1)–(C.4). Part (C.1) convoluted with p _μ gives

(C.5)

For (C.2)–(C.3) we correspondingly have

(C.6)

(C.7)

For (C.4) we have

(C.8)

So the sum (C.5)–(C.8) is zero.