Scholarly article on topic 'Ambitwistor strings and reggeon amplitudes in 
                            N
                            =
                            4
                         SYM'

Ambitwistor strings and reggeon amplitudes in N = 4 SYM Academic research paper on "Physical sciences"

CC BY
0
0
Share paper
Academic journal
Physics Letters B
OECD Field of science
Keywords
{"Ambitwistor strings" / "Super Yang–Mills theory" / "Reggeon amplitudes"}

Abstract of research paper on Physical sciences, author of scientific article — L.V. Bork, A.I. Onishchenko

Abstract We consider the description of reggeon amplitudes (Wilson lines form factors) in N = 4 SYM within the framework of four dimensional ambitwistor string theory. The latter is used to derive scattering equations representation for reggeon amplitudes with multiple reggeized gluons present. It is shown, that corresponding tree-level string correlation function correctly reproduces previously obtained Grassmannian integral representation of reggeon amplitudes in N = 4 SYM.

Academic research paper on topic "Ambitwistor strings and reggeon amplitudes in N = 4 SYM"

JID:PLB AID:33141/SCO Doctopic: Theory

Physics Letters B ••• (••••)

[m5Gv1.3;v1.221; Prn:6/09/2017; 13:42] P.1 (1-1

ELSEVIER

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Ambitwistor strings and reggeon amplitudes in N = 4 SYM

L.V. Borka b, A.I. Onishchenkoc d e

a Institute for Theoretical and Experimental Physics, Moscow, Russia

b The Center for Fundamental and Applied Research, All-Russia Research Institute of Automatics, Moscow, Russia c Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia d Moscow Institute of Physics and Technology (State University), Dolgoprudny, Russia e Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia

A R T I C L E I N F 0

A B S T R A C T

Article history:

Received 11 April 2017

Received in revised form 18 July 2017

Accepted 23 August 2017

Available online xxxx

Editor: N. Lambert

Keywords:

Ambitwistor strings Super Yang-Mills theory Reggeon amplitudes

We consider the description of reggeon amplitudes (Wilson lines form factors) in N = 4 SYM within the framework of four dimensional ambitwistor string theory. The latter is used to derive scattering equations representation for reggeon amplitudes with multiple reggeized gluons present. It is shown, that corresponding tree-level string correlation function correctly reproduces previously obtained Grassmannian integral representation of reggeon amplitudes in N = 4 SYM.

© 2017 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

The behavior of scattering amplitudes in high energy or Regge limit is determined by the positions of singularities of their partial wave amplitudes in the complex angular momentum plane. Already an account for leading pole singularities, so called Regge poles, allows to construct phenomenologically successful models. In particular, to explain the experimentally observed asymptotic rise of total cross-section at high energies the Regge pole with the quantum numbers of the vacuum and even parity - the Pomeron was introduced. Later it was realized that in relativis-tic theory Regge poles should be supplemented with Regge cuts, which could be understood as coming from the exchanges of two or more Regge poles. Next, following the pioneering work of Gri-bov [1] the reggeon field theory describing interactions between various reggeons and physical particles was developed [1-3]. Subsequent development of these ideas within the context of quantum chromodynamics, in particular resummation of leading high energy logarithms (as ln s)n to all orders in strong coupling constant (LLA resummation) with the help of Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [4-8], showed the LLA reggeization of QCD scattering amplitudes. The corresponding Regge pole was identified with reggeized gluon. The latter has quantum numbers of the ordinary gluon and Regge trajectory j(t) passing through

E-mail address: borkleonid@gmail.com (L.V. Bork).

unity at t = 0. Today BFKL equation is known at next-to-leading-logarithmic-approximation (NLLA) [9,10] and the reggeization of QCD amplitudes is also proven at NLLA [11]. In general, the amplitudes with reggeized gluons (also known as gauge invariant offshell amplitudes) [12-17] arise either in the study of multi-Regge kinematics [18-21] or within the context of kT or high-energy factorization [22-25].

On the other hand, recently Roiban, Spradlin and Volovich (RSV) based on Witten's twistor string theory [26] got the description of Nk—2MHVn N = 4 SYM tree level amplitudes in terms of integrals over the moduli space of degree k — 1 curves in super twistor space [27,28]. Subsequent generalization of RSV result by Cachazo, He and Yuan (CHY) led to the discovery of so called scattering equations [29-33]. Within the latter tree level N = 4 SYM amplitudes are written in terms of integrals (localized on the solutions of mentioned scattering equations) over the marked points on the Riemann sphere. Subsequently the CHY formulae together with their loop level generalization (see [34] and references therein) were derived from ambitwistor string theory [35,36].

Another very close research direction is related to the representation of Nk—2MHVn scattering amplitudes in terms of integrals over Grassmannians [37-42]. The latter naturally unifies different BCFW [43,44] representations both for tree level amplitudes and loop level integrands [37,38] and is ultimately connected to the integrable structure behind N = 4 SYM S-matrix [45-49].

http://dx.doi.org/10.1016/j.physletb.2017.08.070

0370-2693/© 2017 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

L.V. Bork, A.I. Onishchenko/ Physics Letters B ••• (••••) •

[m5Gv1.3;v1.221; Prn:6/09/2017; 13:42] P.2(1-8)

The use of mentioned representations (Grassmannian, RSV, scattering equations and so on) of Nk-2MHVn amplitudes (i.e. the full tree level S-matrix of N = 4 SYM) provides us with relatively compact analytical expressions for n-point tree level amplitudes, which in their turn could be used to compute corresponding loop level amplitudes via modern unitarity based methods both at high orders of perturbation theory and/or with large number of external particles in N = 4 SYM and other field theories including QCD (see for a review [50]). It is important to note that these results was almost impossible to obtain by standard Feynman diagram methods.

The aim of the present work is to extend the recently obtained results for the usual N = 4 SYM scattering amplitudes and to derive scattering equations representation for reggeon amplitudes in N = 4 SYM from four dimensional ambitwistor string theory. At a moment, there are already scattering equations representations for the form factors of operators from stress-tensor operator supermultiplet and scalar operators of the form Tr($m) [51,52]. Also some formulae were extended to Standard Model amplitudes [53]. Besides, there are several results for the Grassmannian integral representation of form factors of operators from stress-tensor operator supermultiplet [54-57] and reggeon amplitudes (form factors1 of Wilson line insertions) [58,59], see also [60] for a recent interesting duality for Wilson loop form factors. A very close research direction is the twistor and Lorentz harmonic chiral superspace formulation of form factors and correlation functions developed in [61-67], see the discussion in conclusion.

This paper is organized as follows. First, in section 2 we introduce necessary definitions for reggeon amplitudes (Wilson lines form factors). Next, in section 3 after recalling some of the basic facts of four dimensional ambitwistor string theory we proceed with the construction of string vertex operator for reggeized gluon and derive scattering equations representation for reggeon amplitudes from corresponding string correlation functions. Finally, in section 4 we come with our conclusion.

2. Reggeon amplitudes and Wilson lines

To describe amplitudes with reggeized gluons it is convenient to use the representation of the latter in terms of Wilson line operators as in [15]:

Wp (k) = d xe'

j4„ Jxk

x Tr < — tc P exp

ds p ■ Ab (x + sp)tb

Here tc is SU(Nc) generator,2 k (k2 = 0) is the reggeized gluon momentum and p is its direction or polarization vector, such that p2 = 0, p ■ k = 0. The momentum and polarization vector of the reggeized gluon could be related to each other through so called kT - decomposition of momentum k:

kv = xpv + kV, x e[0,1].

Note, that such decomposition could be also parametrized by an auxiliary light-cone four-vector q^, so that

kV (q) = kv - x(q)pv with x(q) = — and q2 = 0.

' q ■ p

1 See the precise definition in section 2.

2 The color generators are normalized as Tr(tDt6) = i

Using the fact, that the transverse momentum kV is orthogonal to both pv and qv vectors one can decompose it into the basis of two "polarization" vectors3 [12]:

kV (q) = ~

k (p\YV\q] 2 [pq]

with k =

(qp) '

(q\YV\p] 2 (qp)

( p\/\q] ' [pq] .

It is easy to check, that k2 = —kk* and both k and k* variables are independent of auxiliary four-vector q^ [12].

Both usual and color ordered4 reggeon amplitudes with n reggeized and m usual on-shell gluons could be then written in terms of form factors with multiple Wilson line insertions as [15]:

' gm+11... gn+m)

= m, ci, a }m=1 in wpm+j (km+j ) |0>, j=1

here asterisk denotes an off-shell gluon and p, k, c are its direction, momentum and color index. Next ({kj, ci, a 1| =Ç^m=1(ki, £i, a | and (ki, si, ci i denotes on-shell gluon state with momentum ki, polarization vector s- or s+ and color index ci, pi is the direction of the i'th (i = 1, ..., n) off-shell gluon and ki is its off-shell momentum. For the case when only reggeized gluons are present (correlation function of Wilson line operators) we have:

^0+n g... gn) = (0| Wp1 k)... Wpnn (kn)i0>. (6)

In the case of N = 4 SYM we may also consider other on-shell states from N = 4 supermultiplet. The most convenient way to do so is to consider color ordered superamplitudes defined on N = 4 on-shell momentum superspace:

Am+n , gm+1,..., gn+m)

= (£21 ...ßm\[] Wpm+j (km+j )\0),

where ...£m\ ^0i=1(O\£i and £ (i = 1, ..., m) denotes

N = 4 on-shell chiral superfield [69]:

£ = g+ + n A f A + 2 n A n B 0 AB + 31 n A n B n C e abcdÏd

1 ~ ~ ~ ~ ARC D —

+ 4 n a n b n c n d e ABCDg .

Here, g+, g— denote creation/annihilation operators of gluons with + 1 and —1 helicities, ftA, stand for creation/annihilation operators of four Weyl spinors with negative helicity —1/2 and four Weyl spinors with positive helicity correspondingly, while 0AB denote creation/annihilation operators for six scalars (anti-symmetric in the SU(4)r R-symmetry indices AB). All N = 4 SYM fields transform in the adjoint representation of SU(Nc) gauge group. The A*+n (^i,..., g**+m) superamplitude is then the function of the following kinematic5 and Grassmann variables

Am+n (£1,..., gm+n)

(Ui1 ^ i1 m }j=1; {ki ,^p,i1 ^ p,i im+m+1

3 Here we used the helicity spinor decomposition of light-like four-vectors p and q.

4 Here we are dealing with color ordered amplitudes for simplicity. The usual amplitudes are then obtained using color decomposition, see for example [58,68].

5 We used helicity spinor decomposition of on-shell particles momenta.

L.V. Bork, A.l. Onishchenko / Physics Letters B ••• (••••)

[m5Gv1.3;v1.221; Prn:6/09/2017; 13:42] P.3(1-8)

and encodes in addition to amplitudes with gluons also amplitudes with other on-shell states similar to the case of usual on-shell superamplitudes [50].

3. Ambitwistor string correlation functions

As we already mentioned in Introduction our aim here is to derive scattering equations representation for reggeon amplitudes in N = 4 SYM using four dimensional ambitwistor string theory [36], see also [70] for further details. The target space of the latter is given by projective ambitwistor space PA:

PA = {(Z, W) e T x T*| Z • W = °}/{z . 9z _ w . 9w} , (10)

with T and T* denoting twistor and dual twistor spaces. Next, Z = (Xa, fa, xr) e T = C414, W = (f, X, x) e T* and Z • W = Xafa + faXa + XrXr (r = 1, • • •, 4). The corresponding worldsheet theory consists from the worldsheet spinors (Z, W) taking values in T x T* together with GL(1, C) gauge field a serving as a Lagrange multiplier for the constraint Z • W = 0. In the conformal gauge its action is given by

S = — i W • 9Z _ Z • 9 W + aZ • W + SJ,

2n J •"

where d = dada (a, a are some local holomorphic and anti-holomorphic coordinates on Riemann surface E) and Sj denotes the action for the su(Nc) worldsheet Kac-Moody current algebra J e (E, KE ® su(Nc)). KE, as usual, is the canonical bundle on the surface E and the other worldsheet fields take values in6

Z e Q.0(E, KS2 ® T), W e il°(E, KS/2 ® T*), a e Q0,1(E),

To calculate string scattering amplitudes we need vertex operators. There are two equivalent representations for integrated vertex operator used to describe on-shell states [36]:

„a _ [ ^s2(Xa — saX^i^]+Xr^ J ■ Ta , (15)

Va = f — ~S2l4(Xa — saX| fja — saX)es^Xa> J ■ Ta , (16)

where S(z) = d(1/2niz). We would like to stress here, that both vertex operators contain all sixteen on-shell states of N = 4 SYM. To obtain Nk—2MHV on-shell scattering amplitudes one may use for example the correlation function of k V operators and n — k V operators [36]:

Ak,n = V ...VkVk+1 ...Vn) . (17)

This correlation function is not hard to calculate7 and we get [36]:

An,k = /

dsa doa

VolGL(2, C) 1=1 sa(Oa _ Oa+1)

x ]"[ S2(Xp _ spX(Op)) p=k+1 k

x n <52|4(Xi _ siX(Oi), ni _ siX (Oi)),

6 Powers of the canonical bundle denote corresponding field conformal weights.

7 See [36] for details.

x(o) = y , x(o) = y

L--' rr — ri: L--' rr — rr.-

XO) = £

rr —

- O _ Op

p=k+1 p

In terms of homogeneous coordinates on Riemann sphere aa = 1 (1, a) the same result is written as [36]:

An,k =

f_1_n d2°a TÏ

J VolGL(2,C) 1 1 (aa + 1) 11

a=1 p=k+

P(Xp _ X(Op))

xf! ^2|4(X i _ X (Oi ),fji _ X (Oi ))

with (i j) = OiaOa! and

(11) x(o) = £ , X(o) = £

^ (O Oi )

(O Op ) '

x (o) = j2

(OOp )

The scattering equations are then straightforwardly follow from the arguments of delta functions:

ka • P (Oa ) = XaX a P aa (oa ) = X^I^H Xa(Oa )X a (oa ) = 0 •

To describe reggeon amplitudes we also need the ambitwistor string vertex operator for reggeized gluon. The latter could be obtained from the pullback of corresponding ambitwistor space wave function to string theory worldsheet.8 The required ambitwistor space wave function could be easily found using a representation of corresponding reggeon amplitudes with n + 1 legs in terms of convolutions of particle-particle-reggeon PPR vertexes (minimal off-shell amplitudes in the language of [58,59]) with on-shell amplitudes with n + 2 legs.9 This construction also comes naturally [74] by noting that gluing (introducing loop integration) PPR vertex to the on-shell amplitude we get one-loop reggeon amplitude whose leading singularity (extracted by maximally cutting the loop) gives us the corresponding tree level reggeon amplitude.

It should be noted, that this gluing procedure is applicable to any representation of on-shell scattering amplitudes, not only given by ambitwistor string correlation function. In the latter case however the mentioned gluing procedure could be used to reconstruct the ambitwistor string vertex operator for reggeized gluon (Wilson line operator insertion). This way we get

fn+1 d2, .d2jj.

/nVo^ d4f i ^2,2 + 1 (^n ^ g> )

n,n+1 ' M Vol[GL(1)]

x Vn Vn+1

jajb^ifabcjc^jc '

Here, c denotes the color index of the reggeized gluon and we have used projection of tensor product of two adjoint on-shell

8 For the previous work on reggeon string vertexes within superstring theory see [71,72] and references therein.

9 The form of the minimal off-shell vertex could be also obtained from the symmetry arguments along the lines of [73].

L.V. Bork, A.l. Onishchenko/ Physics Letters B ••• (••••)

[m5Gv1.3;v1.221; Prn:6/09/2017; 13:42] P.4 (1-1

gluon color representations onto reggeized gluon adjoint color representation. The minimal PPR vertex A*2 2+1(^n, ^n+;i, g*) is given by [58]: '

A*,2+1(&n, ^n+1, g*)

_ s4(k + XnXn + Xn+1Xn+1) i4 (fjn(pn + 1) + fjn+1(pn)) k* {pn){nn + 1){n + 1 p)

where p = XpXp is the reggeized gluon direction and k * was defined in Section 2 when introducing kT decomposition of the reggeized gluon momentum k. It should be noted, that each of V vertex operators above could be exchanged for V operator and thus the above representation for reggeized gluon vertex operator is not unique. The ambitwistor string vertex operator we got is non-local by construction. The latter property is expected as Wilson line is non-local object by itself. Also, the solution of scattering equations with off-shell leg represented by two on-shell ones (kinematical part) as in (23) does not lead to coordinates of points zn, zn+1 on Riemann sphere being close and hence the OPE expansion of the product of two on-shell vertex operators Vn, Vn+1 as in [71, 72] could be used only in multi-Regge kinematics limit. From the viewpoint of worldsheet theory the Wilson line string vertex operator we got is composite operator build from on-shell vertexes Vn, Vn+1 and is on the same footing as composite operators in any other QFT. Note, that it has the same good worldsheet properties as the simple product of two Vn, Vn+1 operators and the gluing with minimal PPR vertex A* 2+1(Qn, ^^^ g*) may be considered as the corresponding projector or wave function in target projective ambitwistor space PA. Now, performing integrations10 over helicity spinors Xi, Xi we get (the projection operator d4p acting on VnVn+1 is assumed)

WL n,n+1

(ïp> fdfo fdfa 1

01 ^2^2

VnVn+11 ,

2 jQ„ + |jajb^ifabcjc ) jc

Xn = Xn + ,

X n = p1~Xn + (—-J—- Xn+1 ,

- P2 ~

Xn+1 = Xn+1 +--— Xn ,

- P1P2 -

Xn+1 = —fi\Xn+\ — P1P2X ,

fjn = ■

ijn+1 = Pxfoltn ■

Xn = Xp, Xn = ——-, ]2n = np;

- y - (Çp> - y

Xn+1 = X ^ Xn+1 =

Rn+1 = 0,

where X% = (l \ is some arbitrary spinor. In practical calculations it is useful to identify it with the spinor Xq coming from helic-ity spinor decomposition of auxiliary momentum q arising in kT decomposition of reggeized gluon momentum k.

The reggeon amplitude with one reggeized gluon and n on-shell final states is then given by the following ambitwistor string correlation function:

Ak,n+1 = V ...VkVk+1 ...VnVWL1,n+2

See [55,58] for details.

Evaluating first string correlator of on-shell vertexes with the help of (18) we get

(lp) f dft f dfr 1 1

(Ïp> f dft f k * J P2 J

P1 P2 fo VolGL(2, C)

dSadCTa

H S2(Xp — SpX(ap))

=1 Sa (aa — aa+1) p=k\x k

* n 5_2|4(Xi — SiX(ai), ni — SiX(ai)).

Now using unity decomposition as in [41]:

1 _ 1 f dk^(n+2) c VolGL(k) J

x dkxkL (detL)n+2Skx(n+2) (C — L ■ CV[s, a]) ,

where the integral over L matrix is the integral over GL(k) transformations and CV [a] denotes the Veronese map from (C2)n+2/GL(2) to G(k, n + 2) Grassmannian [41] (see also [51]):

Cv [s,a ] =

a V [S1 ,ax] a V [S2,a2]

a V [Sn+2 ,an+2]

a V [s, a] =

\lak—1J where [28,51]: k

li = s—1 H (aj — aj )—1, i e (1, k) j=1,j=i k

li = sif!(aj — ai)-1, i e (k + 1, n + 2).

Next, integrating (30) over sa and aa we get (lp) f f d^1 1 1

k * J P2 J '

k,n+1 k* I A, ! p1 p2 p2 VolGL(k)

* j dk*(n+2)

* s(n+2—k)*2(C!■ X)

j dk*(n+2)c F(C)Skx2(C ■ X)Skx4(C ■ n)

(28) F(C) =

J VolGL

VolGL(2, C)

T+r dSadaa dk*kL Sk*(n+2) (C — L ■ CV[s, a])

1 1 Sa (aa — aa+\) V >

Sk*2(C ■ X ) ^ 52 £ Cai X i) ,

a=1 i=1

n+2 n+2 2

&(n+2—ky*2(C X) = 52 Cj

b=k+1 \j=1 )

JID:PLB AID:33141/SCO Doctopic: Theory

L.V. Bork, A.I. Onishchenko / Physics Letters B ••• (••••) •

[m5Gv1.3;v1.221; Prn:6/09/2017; 13:42] P.5(1-8)

Skx4(C•

n ) ^HE C

Here, C± is the matrix defined by the identity C ■ (C±)T = 0 and it is assumed that all matrix manipulations are performed after GL(k) gauge fixing. Also note that the above delta functions should be thought as S(x) = 1/x and the corresponding contour integral will then compute the residue at x = 0 [75].

By construction F (C) contains (k — 2) x (n — k) delta function factors forcing integral over C matrix to have Veronese form [41]. In general F (C) is a rather complicated rational function of the minors of C matrix. For example, for k = 3 and n + 2 = 6 it is given by [41,76]:

F (C ) = - ,

(123)(345)(561) S

S = (123)(345)(561)(246) - (234)(456)(612)(351).

Here11 (i1 ...ik) is minor of C matrix constructed from i1,..., ik columns of C. The integral over dkx(n+2)C/VolGL(k, C) can then be reduced to multidimensional contour integral over (k — 2)(n — k) complex variables t and evaluated by taking residues. However, at the end after all technicalities (which are highly non-trivial and interesting in their own turn [41,76]) it could be shown that F (C) may be chosen in the form

F (C ) =

(1 ■■■k)(2 •• •k + 1) • • • (n + 2 •••k - 1)

Next, let us rewrite (35) in the form (the proper choice of integration contour ^¡rene+2 is implemented [41,77])

1k,n+1

dkx(n+2)C d01d02

k* j Vol[GL(k)] 0102

kx2 (C' • A) Skx4 (C' • g) S(n+2-k)x2 (C/X •

(1 •••k)(2 •••k + 1) ••• (n + 2 • • k - 1)

Cn+1 = -01 Cn+1 + 0102 Cn+2, 1 + 01,

C— Cx -I-Ln+1 = Ln+1 + 0102

Cn+2 = -01 Cn+2 +

1 + 01

Cn+2 = Cn+2 + 02 Cn+\,

Ai = Ai, i = 1,. .. n, An+1 — Ap,

A i = A i, i = 1,. .. n, An+1 {%\k {%P),

m = f} i, i = 1,. . .n, în+1 = n p,

An+2 = %

An+2 = -

_ _ ¡¡n+2 = 0.

Now, introducing inverse C-matrix transformation

Cn+1 = Cn+1 + 02 C„+2,

Cn+1 + Cn+2,

Cn+2 = + 0 C'„

minors of C-matrix containing both n + 1 and n + 2 columns when rewritten in terms of minors of C'-matrix acquire extra — factor. For example, for (n + 1 k — 2) minor we have

(n + 1 • • • k - 2) = - — (n + 1 • • • k - 2)'. 01

On the other hand, minors containing either n + 1 or n + 2 column transform as

(n + 21 k - 1 )

= l+h (n + 11 • • • k - 1)' + (n + 21 • • • k - 1)', 0102 ( ) ( ) ,

(n - k + 2 • • • n + 1)

= (n - k + 2 • • • n + 1 )' + 02(n - k + 2 • • • nn + 2)',

while all other minors remain unchanged ( ■ ■ ■ ) = ( ■ ■ ■ )'. Now, going to the integral over C' matrix and accounting for the transition

Jacobian ^^^ we get

{% p) f dkx(n+2) C ' d01d02 Xkx2,r ' n kxUr / Vol[GL(k)] 0102

'-Skx2 (C'• A Skx4 (C'• il)

x s(n+2-k)x2^c'X

(1^ • • !<)'• • • (n + 2 • • • k - 1)' (n - k + 2 • • • nn + 2)'

x 1 + 02

x 0102 + (1 + 01 )

(n - k + 2 • • • nn + 1)'

(n + 11 • • • k - 1)'

(n + 21 • • • k - 1)'

Next, taking first residue at j2 = 0 and then at j =—1 (i.e. considering corresponding residual form) we recover our previous result for reggeon amplitude with one reggeized gluon [58] (here we again assume projection operator acting on Grassmannian integral):

' k,n+2

dk x(n+2) C ' Vol[GL(k)]

Reg. =

skx2 C' • D skx4 (c' • f^j s(n+2-k)x2 C• A)

(1 • • • !<)'• • • (n + 1 • • • k - 2)'(n + 2 1 • • • k - 1)'

{%p) (n + 2 1 • • • k - 1)'

k* (n + 11^ • • k - 1)'

We have also verified that direct evaluation of (30) reproduces all particular off-shell amplitudes considered as examples in [58].

Finally we need to perform inverse operation, that is to reduce integral in (47) to the integral over G(2, n + 2) Grassmannian. This is again done with the help of Veronese map [41,51]. Using resolution of unity (31), fixing GL(k) gauge to enforce rational form of scattering equations [36] and performing integration over C matrix our Grassmannian integral representation (47) takes the form of scattering equations representation we are looking for:

11 We hope there will be no confusion with previous definition (i j) = aiaaa used in d2oa integrals over homogeneous coordinates on Riemann sphere before.

12 The Reg. notation is chosen because this ratio of minors regulates soft holomor-phic limit with respect to external kinematical variables associated with reggeized gluon [58].

L.V. Bork, A.l. Onishchenko/ Physics Letters B ••• (••••) •

[m5Gv1.3; v1.221; Prn:6/09/2017; 13:42] P.6(1-8)

(aa +1) VolGL(2,

- H S2(Xp — X(ap)) p=k+1

*n&24(li — X(ai),rn — X (ai)),

i=1 where

V (lp) (kn + 1)

Reg.V =--

k* (kn + 2)

and we have also performed the transition to homogeneous coordinates on Riemann sphere. The doubly underlined functions are defined as

(a ai )

(a ap ) = (a ap )

p=k+1 p p=k+1 v'

. (51)

The result for the case of reggeon amplitudes with multiple reggeized gluons A*+n could be obtained along the same lines. For example, in the case with first m particles on-shell and last n being reggeized gluons we would get:

,, n+2

n+2 d2aa Reg.V(m + 1,...,m + n) m+2n -2(j j(

- H S (Xp — X(ap))

(aa +1)

VolGL(2, C)

*n S2|4(Xi — X (ai ),Ri — | (ai )), i=1

Reg.V (m + 1,...,m + n) = ]"[ RegV.( j + m),

(ïjpj > (k 2 j — 1 + m)

Reg.V (j + m) =

k ** (k 2 j + m) '

and external kinematical variables are defined as

Xi = Xi, i = 1,...m,

Xm+2 j — 1 = Xpj, Xm+2 j = l j, j = 1,...n, Xi = X i, i = 1,...m,

(lj\km+j ~ _ (pj\km+j

Xm+2 j — 1 =

(ïjpj >

^m+2 j = '

(ïjpj > '

j = 1,...n,

fj_i = fj i, i = 1,...m,

U.m+2 j — 1 = ¿j pj, gm+2 j = 0, j = 1,...n. (54)

Note, that it is possible to rewrite (52) as an integral over Gr(k, m + 2n) Grassmannian coinciding with our previous result [59].

At the end of this section we want to make a short comment on how these results could be incorporated into modern approaches to the computation of scattering amplitudes in gauge theories. The insertion of our new VW+1 vertex operator can be reformulated in the language of BCFW recursion as an application of some simple linear integral operator to individual BCFW terms for on-shell scattering amplitudes This should be much more simple approach compared to the use of analog of BCFW recursion for the off-shell gauge invariant amplitudes themselves [12-15]. Next, considered tree level off-shell gauge invariant amplitudes can be further used in some variations of generalized unitarity approach at loop level (see for a review [50]).

4. Conclusion

In this paper we presented results for scattering equations representations for reggeon amplitudes in N = 4 SYM derived from four dimensional ambitwistor string theory. The presented derivation could be also easily generalized to the case of tree level form factors of local operators and loop integrands of reggeon amplitudes, which will be the subject of our forthcoming publication [74].

As by product we found an easy and convenient gluing procedure (linear integral operator) allowing us to obtain required reggeon amplitude expressions from already known on-shell amplitudes. The construction of string vertex operator for reggeized gluon was inspired by the mentioned gluing procedure. This gluing procedure can also be formulated in momentum twistor space which rises an interesting question about its relation to recent developments in the computation of form factors [61-63] within twistor approach to N = 4 SYM [78,79].

It would be extremely interesting to consider pullbacks of composite operators defined on twistor or Lorentz harmonic chiral superspace [61-67] to construct corresponding string vertex operators. We hope that along these lines we will be able to get scattering equations representation for arbitrary local composite operators.

Having obtained scattering equations representations one may wonder what is the most efficient way to get final expressions for amplitudes with given numbers of reggeized gluons and other on-shell states. In the case of usual on-shell amplitudes we know that they could be obtained through the computations of global residues by the methods of computational algebraic geometry [80-82], see also [83]. It would be interesting to see how this procedure works in the case of reggeon amplitudes considered here.

Finally, we should also develop methods for computing loop corrections to reggeon amplitudes together with their loop level generalization of scattering equations representation. Besides, it is extremely interesting to see how the presented approach works in gravity and supergravity theories, where we have a well developed framework for reggeon amplitudes based on high-energy effective lagrangian, see [84,85] and references therein.

Acknowledgements

The authors would like to thank D.I. Kazakov, L.N. Lipatov and Yu-tin Huang for interesting and stimulating discussions. This work was supported by RSF grant #16-12-10306.

References

[1] V.N. Gribov, A Reggeon diagram technique, Sov. Phys. JETP 26 (1968) 414-422, Zh. Eksp. Teor. Fiz. 53 (1967 ) 654.

[2] H.D.I. Abarbanel, J.B. Bronzan, R.L. Sugar, A.R. White, Reggeon field theory: formulation and use, Phys. Rep. 21 (1975) 119-182.

[3] M. Baker, K.A. Ter-Martirosian, Gribov's Reggeon calculus: its physical basis and implications, Phys. Rep. 28 (1976) 1-143.

[4] L.N. Lipatov, Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories, Sov. J. Nucl. Phys. 23 (1976) 338-345, Yad. Fiz. 23

(1976) 642.

[5] E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Multi-Reggeon processes in the Yang-Mills theory, Sov. Phys. JETP 44 (1976) 443-450, Zh. Eksp. Teor. Fiz. 71 (1976) 840.

[6] V.S. Fadin, E.A. Kuraev, L.N. Lipatov, On the Pomeranchuk singularity in asymptotically free theories, Phys. Lett. B 60 (1975) 50-52.

[7] E.A. Kuraev, L.N. Lipatov, V.S. Fadin, The Pomeranchuk singularity in nonabelian gauge theories, Sov. Phys. JETP 45 (1977) 199-204, Zh. Eksp. Teor. Fiz. 72

(1977) 377.

[8] I.I. Balitsky, L.N. Lipatov, The Pomeranchuk singularity in quantum chromody-namics, Sov. J. Nucl. Phys. 28 (1978) 822-829, Yad. Fiz. 28 (1978) 1597.

[9] V.S. Fadin, L.N. Lipatov, BFKL pomeron in the next-to-leading approximation, Phys. Lett. B 429 (1998) 127-134, arXiv:hep-ph/9802290.

JID:PLB AID:33141/SCO Doctopic: Theory

L.V. Bork, A.I Onishchenko / Physics Letters B ••• (••••)

[m5Gv1.3;v1.221; Prn:6/09/2017; 13:42] P.7(1-8)

11 [15

13 [16

15 [17

17 [18

18 [19

20 [20

22 [21

24 [22

26 [23

27 [24

29 [25

33 [27

35 [28

38 [30

40 [31

41 [32

44 [33

46 [34

49 [35

50 [36

52 [37

54 [38

56 [39

59 [40

61 [41

M. Ciafaloni, G. Camici, Energy scale(s) and next-to-leading BFKL equation, [43 Phys. Lett. B 430 (1998) 349-354, arXiv:hep-ph/9803389.

V.S. Fadin, R. Fiore, M.G. Kozlov, A.V. Reznichenko, Proof of the multi-Regge [44 form of QCD amplitudes with gluon exchanges in the NLA, Phys. Lett. B 639 (2006) 74-81, arXiv:hep-ph/0602006.

A. van Hameren, BCFW recursion for off-shell gluons, J. High Energy Phys. 07 [45 (2014) 138, arXiv:1404.7818 [hep-ph].

A. van Hameren, M. Serino, BCFW recursion for TMD parton scattering, J. High Energy Phys. 07 (2015) 010, arXiv:1504.00315 [hep-ph]. [46

K. Kutak, A. Hameren, M. Serino, QCD amplitudes with 2 initial spacelike legs via generalised BCFW recursion, J. High Energy Phys. 02 (2017) 009, arXiv:1611.04380 [hep-ph]. [47

P. Kotko, Wilson lines and gauge invariant off-shell amplitudes, J. High Energy Phys. 07 (2014) 128, arXiv:1403.4824 [hep-ph].

A. van Hameren, P. Kotko, K. Kutak, Multi-gluon helicity amplitudes with one [48 off-shell leg within high energy factorization, J. High Energy Phys. 12 (2012) 029, arXiv:1207.3332 [hep-ph].

A. van Hameren, P. Kotko, K. Kutak, Helicity amplitudes for high-energy scat- [49 tering, J. High Energy Phys. 01 (2013) 078, arXiv:1211.0961 [hep-ph]. L.N. Lipatov, Gauge invariant effective action for high-energy processes in QCD, [50 Nucl. Phys. B 452 (1995) 369-400, arXiv:hep-ph/9502308. [51

L.N. Lipatov, Small x physics in perturbative QCD, Phys. Rep. 286 (1997) 131-198, arXiv:hep-ph/9610276.

R. Kirschner, L.N. Lipatov, L. Szymanowski, Effective action for multi-Regge pro- [52 cesses in QCD, Nucl. Phys. B 425 (1994) 579-594, arXiv:hep-th/9402010. R. Kirschner, L.N. Lipatov, L. Szymanowski, Symmetry properties of the effec- [53 tive action for high-energy scattering in QCD, Phys. Rev. D 51 (1995) 838-855, arXiv:hep-th/9403082. [54

L.V. Gribov, E.M. Levin, M.G. Ryskin, Semihard processes in QCD, Phys. Rep. 100 (1983) 1-150. [55

S. Catani, M. Ciafaloni, F. Hautmann, High-energy factorization and small x heavy flavor production, Nucl. Phys. B 366 (1991) 135-188.

J.C. Collins, R.K. Ellis, Heavy quark production in very high-energy hadron col- [56 lisions, Nucl. Phys. B 360 (1991) 3-30.

S. Catani, F. Hautmann, High-energy factorization and small x deep inelastic scattering beyond leading order, Nucl. Phys. B 427 (1994) 475-524, arXiv:hep- [57 ph/9405388.

E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189-258, arXiv:hep-th/0312171. [58 R. Roiban, M. Spradlin, A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009, arXiv:hep-th/0403190. [59 M. Spradlin, A. Volovich, From twistor string theory to recursion relations, Phys.

Rev. D 80 (2009) 085022, arXiv:0909.0229 [hep-th]. [60

F. Cachazo, S. He, E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (6) (2014) 065001, arXiv:1306.6575 [hep-th]. [61 F. Cachazo, S. He, E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (17) (2014) 171601, arXiv:1307.2199 [hep-th].

F. Cachazo, S. He, E.Y. Yuan, Scattering of massless particles: scalars, gluons and [62 gravitons, J. High Energy Phys. 07 (2014) 033, arXiv:1309.0885 [hep-th]. F. Cachazo, S. He, E.Y. Yuan, Einstein-Yang-Mills scattering amplitudes from scattering equations, J. High Energy Phys. 01 (2015) 121, arXiv:1409.8256 [hep- [63 th].

F. Cachazo, S. He, E.Y. Yuan, Scattering equations and matrices: from Ein- [64 stein to Yang-Mills, DBI and NLSM, J. High Energy Phys. 07 (2015) 149, arXiv:1412.3479 [hep-th].

Y. Geyer, L. Mason, R. Monteiro, P. Tourkine, Two-loop scattering ampli- [65 tudes from the Riemann sphere, Phys. Rev. D 94 (12) (2016) 125029, arXiv:1607.08887 [hep-th].

L. Mason, D. Skinner, Ambitwistor strings and the scattering equations, J. High [66 Energy Phys. 07 (2014) 048, arXiv:1311.2564 [hep-th].

Y. Geyer, A.E. Lipstein, L.J. Mason, Ambitwistor strings in four dimensions, Phys. [67 Rev. Lett. 113 (8) (2014) 081602, arXiv:1404.6219 [hep-th].

N. Arkani-Hamed, F. Cachazo, C. Cheung, J. Kaplan, A duality for the S matrix, [68 J. High Energy Phys. 03 (2010) 020, arXiv:0907.5418 [hep-th]. N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov, J. Trnka, Scattering Amplitudes and the Positive Grassmannian, Cambridge University Press, 2012, arXiv:1212.5605 [hep-th]. [69 N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot, J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, J. High Energy Phys. [70 01 (2011) 041, arXiv:1008.2958 [hep-th].

N. Arkani-Hamed, F. Cachazo, C. Cheung, The Grassmannian origin of dual su-perconformal invariance, J. High Energy Phys. 03 (2010) 036, arXiv:0909.0483 [71 [hep-th].

N. Arkani-Hamed, J. Bourjaily, F. Cachazo, J. Trnka, Unification of residues and [72 Grassmannian dualities, J. High Energy Phys. 01 (2011) 049, arXiv:0912.4912 [hep-th]. [73

L.J. Mason, D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, J. High Energy Phys. 11 (2009) 045, arXiv:0909.0250 [hep- [74 th].

R. Britto, F. Cachazo, B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499-522, arXiv:hep-th/0412308. R. Britto, F. Cachazo, B. Feng, E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602, arXiv:hep-th/0501052.

J.M. Drummond, J.M. Henn, J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, J. High Energy Phys. 05 (2009) 046, arXiv:0902.2987 [hep-th].

J.M. Drummond, J. Henn, G.P. Korchemsky, E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317-374, arXiv:0807.1095 [hep-th].

D. Chicherin, S. Derkachov, R. Kirschner, Yang-Baxter operators and scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 881 (2014) 467-501, arXiv:1309.5748 [hep-th].

R. Frassek, N. Kanning, Y. Ko, M. Staudacher, Bethe ansatz for Yangian invariants: towards super Yang-Mills scattering amplitudes, Nucl. Phys. B 883 (2014) 373-424, arXiv:1312.1693 [math-ph].

N. Beisert, A. Garus, M. Rosso, Yangian symmetry and integrability of planar N = 4 super-Yang-Mills theory, arXiv:1701.09162 [hep-th]. H. Elvang, Y.-t. Huang, Scattering amplitudes, arXiv:1308.1697 [hep-th]. A. Brandhuber, E. Hughes, R. Panerai, B. Spence, G. Travaglini, The connected prescription for form factors in twistor space, J. High Energy Phys. 11 (2016) 143, arXiv:1608.03277 [hep-th].

S. He, Z. Liu, A note on connected formula for form factors, J. High Energy Phys. 12 (2016) 006, arXiv:1608.04306 [hep-th].

S. He, Y. Zhang, Connected formulas for amplitudes in standard model, arXiv:1607.02843 [hep-th].

L.V. Bork, A.I. Onishchenko, On soft theorems and form factors in N = 4 SYM theory, J. High Energy Phys. 12 (2015) 030, arXiv:1506.07551 [hep-th]. R. Frassek, D. Meidinger, D. Nandan, M. Wilhelm, On-shell diagrams, Graßman-nians and integrability for form factors, J. High Energy Phys. 01 (2016) 182, arXiv:1506.08192 [hep-th].

M. Wilhelm, Form Factors and the Dilatation Operator in N = 4 Super Yang-Mills Theory and Its Deformations, PhD thesis, Humboldt U., Berlin, 2016, arXiv:1603.01145 [hep-th].

L.V. Bork, A.I. Onishchenko, Grassmannians and form factors with q2 = 0 in N = 4 SYM theory, J. High Energy Phys. 12 (2016) 076, arXiv:1607.00503 [hep-th].

L.V. Bork, A.I. Onishchenko, Wilson lines, Grassmannians and gauge invariant off-shell amplitudes in N = 4 SYM, arXiv:1607.02320 [hep-th]. L.V. Bork, A.I. Onishchenko, Grassmannian integral for general gauge invariant off-shell amplitudes in N = 4 SYM, arXiv:1610.09693 [hep-th]. D. Chicherin, P. Heslop, G.P. Korchemsky, E. Sokatchev, Wilson loop form factors: a new duality, arXiv:1612.05197 [hep-th].

L. Koster, V. Mitev, M. Staudacher, M. Wilhelm, Composite operators in the twistor formulation of N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 117 (1) (2016) 011601, arXiv:1603.04471 [hep-th]. L. Koster, V. Mitev, M. Staudacher, M. Wilhelm, All tree-level MHV form factors in N = 4 SYM from twistor space, J. High Energy Phys. 06 (2016) 162, arXiv:1604.00012 [hep-th].

L. Koster, V. Mitev, M. Staudacher, M. Wilhelm, On form factors and correlation functions in twistor space, arXiv:1611.08599 [hep-th].

D. Chicherin, E. Sokatchev, N = 4 super-Yang-Mills in LHC superspace part I: classical and quantum theory, J. High Energy Phys. 02 (2017) 062, arXiv:1601.06803 [hep-th].

D. Chicherin, E. Sokatchev, N = 4 super-Yang-Mills in LHC superspace. Part II: Non-chiral correlation functions of the stress-tensor multiplet, J. High Energy Phys. 03 (2017) 048, arXiv:1601.06804 [hep-th].

D. Chicherin, E. Sokatchev, Demystifying the twistor construction of composite operators in N = 4 super-Yang-Mills theory, arXiv:1603.08478 [hep-th]. D. Chicherin, E. Sokatchev, Composite operators and form factors in N = 4 SYM, arXiv:1605.01386 [hep-th].

L.J. Dixon, Calculating scattering amplitudes efficiently, in: QCD and Beyond. Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-95, Boulder, USA, June 4-30, 1995, 1996, pp. 539-584, arXiv:hep-ph/9601359.

V.P. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett. B 214 (1988) 215-218.

Y. Geyer, Ambitwistor Strings: Worldsheet Approaches to Perturbative Quantum Field Theories, PhD thesis, Oxford U., Inst. Math., 2016, arXiv:1610.04525 [hep-th].

M. Ademollo, A. Bellini, M. Ciafaloni, Superstring Regge amplitudes and emission vertices, Phys. Lett. B 223 (1989) 318-324.

M. Ademollo, A. Bellini, M. Ciafaloni, Superstring Regge amplitudes and gravi-ton radiation at Planckian energies, Nucl. Phys. B 338 (1990) 114-142. N. Arkani-Hamed, F. Cachazo, J. Kaplan, What is the simplest quantum field theory?, J. High Energy Phys. 09 (2010) 016, arXiv:0808.1446 [hep-th]. L.V. Bork, A.I. Onishchenko, Four dimensional ambitwistor strings and form factors of local and Wilson line operators, arXiv:1704.04758 [hep-th].

80 81 82

100 101 102

110 111 112

120 121 122

L.V. Bork, A.l. Onishchenko/ Physics Letters B ••• (••••)

[m5Gv1.3;v1.221; Prn:6/09/2017; 13:42] P.8(1-l

1 [75] N. Arkani-Hamed, J. Bourjaily, F. Cachazo, J. Trnka, Local spacetime physics from [80] M. S0gaard, Y. Zhang, Scattering equations and global duality of residues, Phys. 66

2 the Grassmannian, J. High Energy Phys. 01 (2011) 108, arXiv:0912.3249 [hep- Rev. D 93 (10) (2016) 105009, arXiv:1509.08897 [hep-th]. 67

3 th]. [81] J. Bosma, M. S0gaard, Y. Zhang, The polynomial form of the scattering equations 68

4 [76] D. Nandan, A. Volovich, C. Wen, A Grassmannian etude in NMHV minors, is an H-basis, Phys. Rev. D 94 (4) (2016) 041701, arXiv:1605.08431 [hep-th]. 69

J. High Energy Phys. 07 (2010) 061, arXiv:0912.3705 [hep-th]. [82] Y. Zhang, Lecture notes on multi-loop integral reduction and applied algebraic

5 [77] L. Dolan, P. Goddard, Gluon tree amplitudes in open twistor string theory, geometry, arXiv:1612.02249 [hep-th], 2016. 70

6 J. High Energy Phys. 12 (2009) 032, arXiv:0909.0499 [hep-th]. [83] S. Weinzierl, Tales of 1001 gluons, arXiv:1610.05318 [hep-th], 2016. 71

7 [78] L.J. Mason, D. Skinner, The complete planar S-matrix of N = 4 SYM as a Wil- [84] L.N. Lipatov, Effective action for the Regge processes in gravity, Phys. Part. Nucl. 72

8 son loop in twistor space, J. High Energy Phys. 12 (2010) 018, arXiv:1009.2225 44 (2013) 391-413, arXiv:1105.3127 [hep-th]. 73 [hep-th]. [85] L.N. Lipatov, Euler-Lagrange equations for the Gribov reggeon calculus in QCD

9 [79] T. Adamo, M. Bullimore, L. Mason, D. Skinner, Scattering amplitudes and Wilson and in gravity, Int. J. Mod. Phys. A 31 (28-29) (2016) 1645011. 74

10 loops in twistor space, J. Phys. A 44 (2011) 454008, arXiv:1104.2890 [hep-th]. 75

35 100

36 101

37 102

38 103

39 104

40 105

41 106

42 107

43 108

44 109

45 110

46 111

47 112

48 113

49 114

50 115

51 116

52 117

53 118

54 119

55 120

56 121

57 122

58 123

59 124

60 125

61 126

62 127

63 128

64 129

65 130