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Nuclear Physics B 911 (2016) 517-562
www.elsevier.com/locate/nuclphysb
Towards NMHV amplitudes at strong coupling
A.V. Belitsky
Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA Received 24 May 2016; received in revised form 12 August 2016; accepted 13 August 2016 Available online 18 August 2016 Editor: Stephan Stieberger
Abstract
Pentagon Operator Product Expansion provides a non-perturbative framework for analysis of scattering amplitudes in planar maximally supersymmetric gauge theory building up on their duality to null polygonal superWilson loop and integrability. In this paper, we construct a systematic expansion for the main ingredients of the formalism, i.e., pentagons, at large 't Hooft coupling as a power series in its inverse value. The calculations are tested against relations provided by the so-called Descent Equation which mixes transitions at different perturbative orders. We use leading order results to have a first glimpse into the structure of scattering amplitude at NMHV level at strong coupling.
© 2016 The Author. 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 formulation of the Pentagon [1] Operator Product Expansion [2] for superWilson loop WN on a null polygonal contour paved a way for unraveling analytical structure of scattering superamplitude AN of N particles, they are dual to , at any value of 't Hooft coupling in planar maximally supersymmetric Yang-Mills theory. This convergent series is developed in terms of elementary excitations f of the color flux-tube propagating on the world-sheet stretched on the loop [1],
E-mail address: andrei.belitsky@asu.edu. http://dx.doi.org/10.1016/j.nuclphysb.2016.08.016
0550-3213/© 2016 The Author. 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.
WN = {0\V\fi)e-tiE*i +°1P*1 +'<Pimi (f1\V\f2)e-t2E*2+°2P*2 +l<hm2
fl,—,f3N-15
...(f3N-15\P\0) , (1.1)
with (N — 5) sets of three conformal cross ratios t, a, <p which encode the shape of the boundary. In addition to the dispersion relations Ef = Ef(u), pf = pf (u) (parametrized by the rapidity u) for f, which were known to all loops for quite some time [9], their form factor couplings to the contour (0\V\fi) and transitions amplitudes (fi\V\fj) between adjacent squares in a geometric tessellation of the polygon were uncovered in a series of recent papers [10-18].
For a few notable exceptions [1,12,19,20], recent literature was predominantly focused on perturbative analyses of scattering amplitudes at weak coupling where a plethora of data is available from different formalisms such as hexagon [21,22] and heptagon [23-26] bootstraps. The reason for this is that at each order in 't Hooft coupling there is only a very small number of flux-tube excitations which determine the amplitude in question. At strong coupling on the contrary, summation over their infinite number should be performed to reproduce the minimal area result obtained within the Thermodynamic Bethe Ansatz [3,27,28] as well as to systematically decode all higher order corrections in 1 /g. At leading order in the inverse coupling, this was effectively demonstrated recently in Ref. [20] following the route outlined in Ref. [12] for MHV amplitudes. Since the dominant contribution at infinite coupling is essentially insensitive to the helicity of external particles involved in scattering, one anticipates to find a factorized overall minimal area prefactor in non-MHV amplitudes as well [29].
In this work, we initiate a systematic study of the flux-tube pentagons in the perturbative regime at strong coupling. Presently, we will not attempt however to unravel the structure of nonperturbative e-ng corrections, though these can be systematically accounted for upon a more thorough consideration. They will become important in the analysis of the transition region from strong to finite and then weak coupling. Compared to other excitations,—fermions, gluons and bound states thereof,—scalars, also known as holes, possess exponentially vanishing masses at strong coupling, potentially producing leading order contribution in the multi-collinear kinematics, i.e., Ti ^ to. However, their effect in the amplitude is formally suppressed by inverse coupling relative to semiclassical string effects and, for this reason, we will ignore holes in the nonperturbative regime, though they were shown to provide an additive geometry-independent constant contribution to the minimal area due to their intricate infrared dynamics when resummed to all orders [19]. We further comment on their contribution to NMHV amplitudes in the Conclusions.
For the exception of the scalars, which are not presently discussed, the perturbative string regime corresponds to the one where the rapidity of excitations scales with 't Hooft constant, u = 2gU as g is sent to infinity while U is kept fixed. For the gauge fields and bound states, the physical region of u corresponds to the interval (—1, 1), while for fermions, u resides on the small fermion sheet containing the point of the fermion at rest and thus varies over two semi-infinite segments u e (—to, —1) U (1, to). It is for these values, the energy E and momentum p of these flux-tube excitations are of order one in g, i.e., E, p ~ g0. Elsewhere, we are entering kinematics where they scale as a power of g (fractional for near-flat or integer for semiclassical regimes) yielding exponentially suppressed contribution to the Wilson loop.
Our subsequent consideration is organized as follows. In the next section, we start with small fermions and solve their parity even and odd flux-tube equations in inverse powers of the coupling. These are used then to construct direct and mirror S-matrices which enter as the main ingredients into the fermion-fermion pentagons. We continue in Sect. 3 with a similar consider-
ation for gluons and their bound states. Due to complicated analytical structure on the physical sheet, we first pass to the half-mirror, or Goldstone, sheets and perform the strong coupling analysis there. In this manner, we find bound-state-bound-states pentagons. In Sect. 4, we use the results of the previous two sections to find mixed fermion-bound-states pentagons verifying consistency of our findings by means of the exchange relations. Another layer of consistency checks arises from consequences of the Descent Equation for superamplitudes in Sect. 5. Finally, we apply our construction to resum the entire series of gluon bound states and effective fermion-antifermion strong-coupling bound pairs for a particular component of NMHV amplitude, observing anticipated factorization of the minimal area from a helicity-dependent prefactor. Finally, we conclude. Several appendices contain compendium of integrals needed for calculations involved as well as a list of results which are two cumbersome to be quoted in the main text.
2. Small fermion transitions
Let us start our consideration with fermions. As we advertised in the Introduction, only the fermion living on the small Riemann sheet survives at strong coupling. In the theory of the flux-tube, the direct and mirror scattering matrices for small-fermion-small-(anti)fermion elementary excitations (and consequently their pentagon transitions) [12,13]
Sff(u, v) = Sff(u, v)
= exp (-2i/f(f1)(u, v) + 2iff (u, v)) , (2.1)
u — v + i
S*ff(u, v) =-Srff (u, v)
u — v
= exp f3(u,v) — 2f(4)(u, v)) , (2.2)
are determined by means of the dynamical phases
f^}(u,v) = —2j Y cos(vt)vl u(2gt), f(2)(u,v) = —2j d sin(vt)Y— u(2gt), 0 0
f(p(u,v) = +2j d sin(vt)y—,u(2gt), f(4)(u,v) = —2j d cos(vt)Y+,u(2gt), 00
which depend on the solutions to the flux-tube equations with sources specific to the type of excitation under consideration. Let us turn to the solution of the u -parity even and odd functions Y— u and y— u, respectively, at strong coupling.
2.1. General solution for even u-parity
An infinite set of even u-parity flux-tube equations for the small fermion [9,13] can be cast in the form1
1 This is achieved by means of the Jacobi-Anger summation formulas, and differentiation w.r.t. the rapidity v.
dt sin(vt)
dt cos(vt)
Y+ ,u(2gt) Y- ,u(2gt)
1 - e-
et - 1
dt sin(vt) cos(ut) ,
Y- uu(2gt) + Y+ ,u(2gt)
1 - e-
et - 1
These are valid for |u| < 2g, while the rapidity of the small fermion resides in the domain |u| > 2g. As was demonstrated in Refs. [30,31], the above equations, or rather their analogues for the ground state of the flux tube,—the cusp anomalous dimension,—can be significantly simplified by performing a transformation to
ru (t) = r+u (T) + ¿r-uu(T) = [1 + i coth - ) yU(T) where we introduced a complex flux-tube function
yU(t) = Y+ uu(T) + ¿Y- ,U(t). In this way the integrands in the left-hand side read
Y+ ,u(2gt) Y- uu(2gt)
1 - e-t et - 1 2
Y-,u(2gt) Ylu&gV _ 1 1 - e-t + et - 1 2
r+uu(2gt) + r-uu(2gt)
r-uu(2gt) - r+uu(2gt)
(2.9) (2.10)
By rescaling the rapidity U = u/(2g) and the integration variable t = 2gt, the equations cease to possess explicit dependence on the 't Hooft coupling. The dependence on the latter is induced however via analyticity conditions on their solutions as will be done in Sect. 2.3.
To solve the above equations it is instructive to first reduce them to a singular integral equation by means of a Fourier transformation [31]. Namely, we introduce the Fourier transforms of the functions involved
ÇU(p) =/ 2dTeiPTYU(T), <(p) = f dne'PTrU(t) .
(2.11)
Notice that since yU(t) is an analytic function in the complex plane, it admits a convergent expansion in terms of Bessel functions, i.e., y(t) ~ y.n jn(t), on the real axis. Thus the support of its Fourier transform is restricted to the interval |p| < 1, i.e.,
vU (P)||p|>1 = 0 >
(2.12)
while ®u(p) is nonvanishing on the entire real line. The inverse Fourier transform of the latter can be decomposed in terms of t-even and odd functions
r+,u(T) = I dp cos(pT)$U(p), r-uu(T) = - / dp sin(pT)$U(p),
(2.13)
respectively, which due to the fact that the function $>U is real, [$U(t)1* = $U(t), correspond to the real and imaginary part of rU.
Now we are in a position to derive an integral equation for the Fourier transform $U. To this end, we replace the two linear combinations (2.9) and (2.10) by their right-hand sides in the flux-tube equations (2.5) and (2.6), respectively, and rescale the integration variable and rapidities as explained after Eq. (2.10). Next, we substitute the definitions (2.13) into the equations obtained in the previous step and evaluate the emerging t -integrals which result in simple rational functions of rapidities. Adding up the two results, we find that $U obeys the following equation
f f dk f <(P) + j —fk)
dk f V
k — p 1
= — dk&uik)
f o(k2 — 1) 1
k — U 2n
p — u p + u
= Ju (P), (2.14)
where we changed the variable v to v = p and the integral is defined by means of the Cauchy principal value P. Due to the original domain of the validity of (2.5) and (2.6), this equation defines $U(p) for |p| < 1 only. However, the inhomogeneity on its right-hand side involves $UP) outside of the interval (-1, 1). This contribution can be found by noticing that due to the support region of q>f, its Fourier transform behaves as yfT ~ e|T 1 for large complex t . As a consequence, the integral (2.11) for $U (p) can be computed by means of the Cauchy theorem (with residues emerging from the trigonometric prefactor) by closing the contour at infinity. This can be done however only provided |p| > 1, yielding
<(P)\\P\>1 = 0(p — 1)J2ciu + (n g)e—4nng(p—1)
+ 0(—p — 1) Y, ciu — (n, g)e—4nng(—p—1), (2.15)
c±(n, g) = T4gYu(±i4ngn)e—4nng . (2.16)
Obviously it is nonperturbative in its origin and still involves unknown coe fficients in its decomposition. They will be fixed in Sect. 2.3.
Summarizing, the solution to the flux-tube equation (2.14) can be rewritten as a sum of solutions to homogeneous and inhomogeneous equations following standard methods [32]
<(p> = — (1+p r+1 f (p) - (1+p V'4 f dk (i-i Y'4 -P- JU k,.
u(p) p + H1 - p) 2 \1 - p) J 2n\ 1 + kj k - p u( )
(2.17)
One can easily verify a posteriori that this indeed solves (2.14) making use of integrals (A.14) and (A.15). Substituting the source Jf, and partitioning the denominator, we can evaluate the resulting integrals making use of Eqs. (A.14) and (A.15), such that
(1 + p\!/4 f dk 0(k2 — 1) fk — A1/4 f
"u(p)\\p\<1=*u(p)—nr—p) J ^-WikTi) $u(k)'
4 (p) =
i + p 2«j2n\ 1 - P
p - u \ û + 1
û - 1
û + 1
p + û \ û - 1
(2.19)
and where in the last term of (2.18) one has to substitute the expansion (2.15) and the constant c was set to zero to comply with properties of scattering phases. Fourier transforming back, we find the all-order expression for rU (t),
f (t) = xU (t) + £ Zgu—T [—irV0(—ir)Uï (4ngn) + 4ngnV1(—ix)U-(4xgn)]
f,+( )
+ H [—iTV0(—iT)U+ (4ngn) + 4ngnV1(-iT)U0+(4ngn)] ,
4ngn + iT
XU(T) = -
1 / û - 1
4 V û + 1
W(-iT,û) --
1 ( u + 1
4 V m - 1
W(-iT, -w).
(2.20)
(2.21)
The integral representations of the special functions involved are given in Appendix A. We will turn to defining the expansion coefficients after we address the u-parity odd case first in the next section.
2.2. General solution for odd u -parity
Up to minor modification, the odd u-parity case is analyzed in a similar manner. Starting with the flux-tube equations [9,13]
dt sin(vt)
9+ iû(2gt) + 9- u(2gt)
1 - e-
et - 1
(2.22)
dt cos(vt)
9- ,u(2gt) 9+ ,u(2gt)
1 - e-
et - 1
dt cos(vt) sin(ut),
(2.23)
we introduce a complex function y via the equation analogous to (2.8) that differs by a relative minus sign
yU(t) = if u(T) - i Y- ,u(T), and pass to a new function
9i (T) = 9+(t) - if-(t) = + i coth 4T^j 9û
such that
9+ ,u(2gt) K-,u(2gt) ^ 1
1 - e-t + et - 1 2
9- ,u(2gt) 9+ ,u(2gt) 1
1 - e-
et - 1
f+,u(2gt) - f-,u(2gt)
f+,u(2gt) + f-,u(2gt)
(2.24)
(2.25)
(2.26) (2.27)
Introducing the Fourier transforms identical to Eqs. (2.11), with however a sign difference for the t-odd part,
r+uu(t) = f dk cos(kT) $U(k), r-uu(t) = f dk sin(kT)fk), (2.28)
we obtain the singular integral equation that $ obeys 1
~f f dk ~f P
<(p) + J ~<(k)
k— p 1
f 0(k2 — 1) 1 = — / dk¥v(k) \ „ ) — — k — u 2n
p — u p + u
(2.29)
As before, we split the solution $v(p) into two regions, the interior of the interval (-1, 1) and its outside. The latter admits an infinite series representation
$U(p)||p|> 1 = 0(p - 1) Ecfu + (n, g)e-4nng(p-1) «>1
+ 0(—p — 1) £ f — (n, g)e—4nng(—p—1), (2.30)
with the expansion coefficients
Cu±(n, g) = T4gyu(±i4ngn)e—4nng , (2.31)
which will be fixed in the next section. Making use of the explicit sources, the solution to Eq. (2.29) yields the function inside the interval (—1, 1),
f ~f r ( 1 + p\1/4 7 dk-(k2 — 1) (k — 1 \1/4~f
=^u(p)—^(i—p) j ^-T—pry—1) fk), (232)
^ (u—1 V/4 _ (uu±1 V/4
p — u \ u ± 1J p ± u \ u — 1/
~f 1 (1 ± ^1/4 b'u (p) = —■ ' p
(2.33)
2*f2n\ 1 - p,
Fourier transforming back, it immediately produces the all-order expression for rU (t),
PU(t) = xU(t) + E CU^T [-iTVo(-iT)U-(4xgn) + 4ngnVl(-iT)U-(4ngn)}
+ E Z n + l [-iTVo(-iT)U+(4ngn) + 4ngnVl(-iT)U+ (4ngn)} ,
(2.34)
f 1 (u-1 y4 „ 1 (u +1 y4
Xu (t) = -{ ^ W(-iT,U) + - ^ W(-iT, -U). (2.35)
4 \ u + 1J 4 \ u - 1/
Now we are in a position to construct a quantization condition for the unknown coefficients cf f,
as well as cu, from the previous section.
2.3. Quantization conditions and their solutions
According to their definitions (2.7) and (2.25), rU (t) and rU (t), respectively, possess an infinite number of fixed zeroes on the imaginary axis at t = 4nigxm due to the trigonometric multiplier present in both, i.e.,
rU (4nigxm) = 0, rU (4nigxm) = 0, (2.36)
with xm = (m - 4 ) where m e Z. They define quantization conditions for the expansion coefficients c±. These can be cast in the explicit form
xf (4nigXm)
V0(4ngXm)
= Y! cU' (n,g)
+ E cfu + (n,g)
XmU- (4ngn) + nr(4ngXm)U0 (4ngn) n + m
XmU+(4ngn) + nr(4ngXm)U+(4ngn)
(2.37)
where we divided both sides by V0 and introduced the ratio
r(z) = -1-). (2.38)
Similar relation holds for c±, where one has to dress everything with tildes. An equation analogous to (2.37), but for the vacuum state describing the cusp anomalous dimension, was proposed and solved in Ref. [3 1 ]. Here we will adopt the strategy advocated there and expand Eq. (2.37) systematically in the inverse powers of the 't Hooft coupling.
Making use of the asymptotic expansion of the special functions for their large argument as given in Appendix A, the above quantization conditions split into two depending on the sign of xm since the functions involved enjoy different asymptotic behavior subject to the condition xm ^ 0. Then the parity-even expansion coefficients admit the form
ciu±(n,g) = (8ngn)±1/4
with explicit a and b being
bfl±(n) 4ng
+ O(1/g2)
bf+(n) =
(n) = -
2 r(n + 4 ) r(n + 1)r2( 4) 2r(n + 4 ) r(n + 1)r2( 4 )
---ln2
Xo+ (u),
--+ - ln2
16 + 8
2Uu (n) = -
x0 (u)
r(n + 4 )
2r(n + 1)r2( 4 )
(2.39)
Xq'-(u), (2.40)
(2.41)
(xQ'+ (u) - 8xÎ0+(u)) + ^ (3Xof'+(u) - 32x[0+ (u))
bf-(n) = -
2r(n + 1)r2 ( 4 )
---+ - ln2
16 + 8
x0o+(u)
(2.42)
--+ - ln2
16 + 8
(x0' (u) - 8x[0 (u)) + ^ (5x0' (u) - 32x10 (u)j
respectively. Here, we introduced inhomogeneities arising from the large coupling expansion of the left-hand side of the quantization condition
xu (±4nig\xm\)
f+ 1 Xf,± (u) 2
= Xio±(u) ±— ± O(1/g2),
Vo(±4ng\Xm\) 0 4ng Xm
with explicit order-by-order contributions being
f,±( ) 1 Xo, (u) = — 4
u ± 1
u ± 1 v u — 1
u^ 1 \ u ± 1
xf{±(u) = —
(u ± 1)2
± 1 y/4 - ±
f,—( ) 5 xl (u)=16
u ± 1
(u — 1)2 V u — 1
u — 1\1/4 (u — 1)2 V u± 1j
u — 1
(u ± 1)2 \ u ± 1
In complete analogy, the solutions to the parity-odd equation read
(n,g) = (8ngn)±l/4
aV±(n) ± ± o(1/g2)
where the a and b coefficients are
au (n) = — „, Xo
r(n ± 1)r2(i)
2lY(n ± 4 )
bu±(n) = ,
u t(- ± 1)r2 ( 4 )
--± - ln2
au (n) = —
x0 (u)
•F(n± 3) _f,—
2T(n ± 1)T2(4)
Xo (u)-
---ln2
fu — 8xfo±(u)) ± ^ (nO^(u) — 32xfo±(u))
bu—(n) = —
'T(n± 3)
2r(n ± 1)T2(4)
---± - ln2
Xo± (u)
--± - ln2
X (u) — 8x[o (u)) ± (5,X0, (u) — 32jf[o (u)j
(2.43)
(2.44)
(2.45)
(2.46)
(2.47)
(2.48)
(2.49)
(2.5o)
respectively, determined by another set of inhomogeneities arising in the left-hand side of the quantization condition
xl (±4nig\xm\) _ f
Vo(±4ng\Xm\)
XXo±(u) =
,'+ 1 X]f,± (u) 2
= Xo±(u) ±— ^ul ± o(1/g2),
u ± 1 v u — 1
4ng Xm
u± 1V/4 1
u^ 1 \ u ± 1
X1±(u) = 16
~f, — f ^ 5 xi (u) = — T6
u — 1
u± n1/4
(u± 1)2 V u — 1) (u — 1)2\ u± 1
u± 1 V/4 1 ( u — n1/4
(u — 1)2 v u — 1
(u ± 1)2 \ u ± 1
(2.51)
(2.52)
(2.53)
(2.54)
The strong coupling expansion can be performed in a straightforward fashion to any required order. To save space we will not present subleading terms explicitly here.
2.4. Strong coupling expansion
Having determined the last unknown ingredients of the solutions, we can sum-up the infinite series in Eqs. (2.34), (2.34) and determine the inverse coupling expansion of the flux-tube functions r and r. For further use, let us decompose the latter in terms of even and odd components with respect to t . They are
±'U(T) = T4
1 / U - 1
w±(t,u) t-
1 ( U + 1
4 V U - 1
W±(t, -U)
(2.55)
X0' (u)
x [V±(t) t 4tVqt(t)] + O(1/g2).
xQ'+(u) in
- + - ln2 V±(t) ± 8 4 / 1 ( ) 4ng
---ln2
rf At) = -
1 / U - 1
4 V U + 1
W ±(T,ii ) + ■
1 / U + 1
W±(t, -U)
(2.56)
xQ' (u)
n 3 \ _i_ xX0'+(u^n 3
- ^ ln2 V,±(t) + 0 ---ln2
8 4 J 1 ( ) 4ng \8 4
x [Vi±(T) T 4tV0^(t)] + O(1/g2),
where we introduced t-even and -odd functions by decomposing W(-it, u) as W(-iT, U) = W+ (t, u) - iW-(T, u) and similarly for V„, see Eqs. (A.12) and (A.13).
The 1 /g expansion of the dynamical phases for the direct and mirror scattering matrices is now preformed in a straightforward fashion by trading y 's for the linear combination of r's according to the equations
Y± fu(T) =
1 ±'U
(t) ± coth4grtf(t)
1 + coth
2 _L 4g
Y± ,u(T) =
±,u(T) T coth4g
rT At)
1 + coth
(2.57)
and expanding the integrands of (2.3) and (2.4) for fixed t . Substituting the above solutions into the scattering phases, we find
fffa)(u1,u2) = j Aïï'(U1, U2) + 4g
- . . 3ln2 (a) ^
Bff (U1, U2) + —— Cf '(U1, U2) 2n
+ O(1/g2 )
(2.58)
(a = 1, 2, 3, 4) with explicit functions deferred to Appendix C.1 due to their length. We verified their correctness by means of the exchange relations that imply that /ff2)(u 1, u2) = /ff1)(u2, ui) as well as symmetry of the mirror phases /fp(u1, u2) = /(3)(u2, u1) and /ff4)(u1, u2) = /f(4)(u2, u1). Further checks will be performed below.
At leading order, the fermion-fermion S-matrix and its mirror read
i U u±-± y4 (ui + i V'4
8g(U 1 -U2) \U 1 + 1/ \U2 - 1/
ln Sff (ui,u2) = -
ui + l\1/4(U2 - n1/4 ' . Ui - W \U2 + 1 !
(2.59)
ln Sf (ui,u2) =
8g(ui - u2)
1 \ 1/4 / - I 1 \ 1/4 ui - i \ I u2 + 1x
ui + 1
u2 - 1
ui + n1/Vu2 - A 1/4 + 2; ui - w V u2 +1)
(2.60)
with fermion-antifermion related to them via Eqs. (2.1) and (2.2). They agree with earlier results of [33]. While the subleading terms are new.
The small-fermion-small-(anti)fermion pentagons
p , . , ¿g -xf[ui]jf[u2]) ^ , . ,
"f|f(ui|u2) =-^-7—;-Pf\f(ui\u2)
2g(ui - u2) iy/1 -xf[ui]Xf[u2] 2g(ui - u2)
(2.61)
exp (-if(p(ui,u2) + iff2(ui,u2) - fff3)(ui,u2)
and the measure read
ßf(u) = -
+ ff4)(ui,u2)) ,
ex^ f(\u, u) - ff4)(u, u)^ ,
(2.62)
i - xf2 [u ]
in terms of the found phases (2.58). Here we introduced a more natural from the point of view of fermions small fermion Zhukowski variable xf[u] = 1 (u — ^u2 — (2g)2) rescaled with the 't Hooft coupling xf[u] = gXf[u ]
= u - Vu2 - i.
(2.63)
To avoid repetitious formulas, we will not display the 1/g expansion of pentagons explicitly which merely reduces to the substitution of Eq. (2.58) with (C.1)-(C.9) into the above formulas, however, we write down the measure to the O(1 /g2) order, which requires taking a limit,
n ± 12ln2(u2 ± 1)"
ßf(u) = -
i - xf2 [u ]
i6g u2 - i
16^g(u2 - i)
+ O(i/g3)
(2.64)
3. Gluon transitions
Now we are turning to the gauge fields and their bound states. The direct and mirror S-matrices for opposite and like helicity gluon stacks can be constructed by a fusion procedure, as was previously reported in Ref. [14]. To avoid complications in algebra due to presence of an infinite number of cuts on the physical sheet, it was instructive to pass to the Goldstone sheet [36], which
is half-way between the real and mirror kinematics. The result of the analysis is summarized in the equations
St1l2 (û1,U2) = S-t2 (û1,U2)SÎ1Î2 (û1,U2) (3.1)
= exp (i2iat1t2(u 1 ,U2) - 2i/t(11t)2(u 1 ,U2) + 2;/^(u 1 ,U2^ , S*i1i2 (u1,u2) = S*t2t (U2,U1) (3.2)
= s*e1 i2 (u1,u2) exp(2at1t2 (u1,u2) + 2 ft (u1,u2) - 2 ft (u1,u2^ , respectively. Here the rational prefactor for the same-helicity S-matrix is
St1t2 (u1,u2) =
f (1 + ^1+2 - i
i'u1 + iu^j f - iu1 + iu2^
f (1 + * + t2 + iu1 - iu2^ f + iu1 - iu-^j
f (1 + t1
-t2 , • + - i
A+t2 I • • \
1 2 2 + iu1 - iu2 I
r (1 + 2 - if1 + iU2) - iu 1 + iU2)
and corresponds to the scattering phase of spin-- magnons for compact XXX spin chain. While in the mirror matrix, it takes the form
s*t1ë2 (u 1 ,u2) = (-1)
f + il 2- iu1 + iu^j f + iu1 - iu2j
f (1 + ^1+2 - iu1 + lm) f
■j^2 + iu1 - iu2j
The dynamical phases in the above equations, in a form slightly different compared to Ref. [14], read for direct
Ol1l2 (u1,u2) =
t(et - 1)
e-i1t/2 sin(u1t)Jo(2gt) - e-i2t/2 sin(u2t) Jo(2gt) (3.5)
- e-№+^)t/2sin ((u1 - u2)t)
/$2 (u1,u2) = / —e-l1t/2sin(u1t)
-ht/2^
'y-,u2 (2gt) + Y+ ,u2 (2gt)
1 - e-
et - 1
/¡%(u1,u2) = j d (e-l1t/2cos(u1t) - Jo(2gt))
9+,u2 (2gt) + 9-,u2 (2gt)
1 - e-
et - 1
and mirror cases
ott (u1,u2) =
t(1 - e-t)
e-t1t/2cos(u1t)Jo(2gt) + e-t2t/2 cos(u2t)Jo(2gt) (3.8) - e-(t1+t2)t/2cos ((u 1 - u2)t) - J^(2gt)
/g2 (u1,u2) = -J ye-t1 t/2sin(u1t)
9-,u2 (2gt) 9g,u2 (2gt)
1 - e-
et - 1
(uiU = + f d (e-tit/2cos(uit) - Jo(2gt}) 0
(3.10)
Though it is not obvious from the above representation, the exchange relations [37,13] imply certain symmetry properties of the dynamical phases. Namely, under the permutation of arguments (and spin labels t), they change as
f^2 (ui,U2) = 4J (U2 ,ui), f(H2 (U1,U2) = f^ (U2 ,ui),
f(42 (U1,U2) = ff4tx (U2 ,Ui). (3.11)
These will be used below as a verification of results obtained at strong coupling. The above expression are well suited to the current strong-coupling analysis, however, we have to transform them first.
Y+,u2 (2gt) Y-,u2 (2gt) 1 - e-t et - 1
3.1. Passing to Goldstone sheet
As we just mentioned above, the physical sheet in the complex u plane possesses an infinite number of cuts [—2g, 2g] stacked up with the interval i along the imaginary axis. For t-gluon bound state, they start from pm[u]| = t/2 and go up/downwards. In the strong-coupling limit, one immediately finds oneself in a predicament, since all of the cuts collapse into one on the real axis pinching the physical region of rapidities —2g < u < 2g. To overcome this complication one has to stay in the latter region but keep away from all of the cuts. This is possible provided one passes to the Goldstone sheet by moving upwards through the first Zhukowski cut in the upper half-plane of u. A distinguished feature of this sheet is that it has only a finite number of cuts ranging from -t/2 to t/2. So one can safely navigate away from [—2g + it/2,2g + it/2] to Sm[u] > t/2 still staying in the strip. When on the Goldstone sheet, one takes the strong coupling limit, and then one can always undo the analytic continuation afterwards and go back to the physical sheet.
According to this discussion, we perform the analytic continuation u -— u + it/2 + i0+ — ug = u for |u| < 2g and immediately find for the flux-tube equations of even
dt sin(vt)
'YG,u(2gt) YGu(2gt)
1 - e-
et - 1
1 f dt sin(vt)""" 2 eiut+t/2,
2 J v sinh 2
(3.12)
dt cos(vt)
YGu(2gt) + YGu(2gt)
1 - e-
et - 1
sinh ^
I dt cos(vt) °'nh 2 eiut-t/2
I oi-TlV» I
(3.13)
and odd parity
dt sin(vt)
dt cos(vt)
Y+,u(2gt) + YG,u(2gt)
1 - e-
1 - e-
et - 1
YGu(2gt) ?G,u(2gt)
et - 1
1 f sinh ^ tn
— dt sin(vt)-2-e /2,
2^ sinh 2
c ft 1 f sinhf iut,tn
--— dt cos(vt)-2- e!Ut+t/2,
2^ sinh 2
(3.14)
(3.15)
respectively, in agreement with Ref. [14]. Again we repeat that these are valid for |u| < 2g and Sm[«j > 2. Notice that the sources are now complex. This will lead to minor differences in the analysis that follows.
The stack-(anti)stack S-matrix with both rapidities on the Goldstone sheet then reads
SGG(u1,u2) = st1t2 (Ui,U2)SgG(UI,U2)
= shi2 (ui,U2) exp (-2ifG^Gi(ui,U2) + 2f/GG(ui,U2)^)
S*GG(u1,u2) = S^GG(u2'U1)
= s*e1 i2 (u 1 ,u2) exp f(ui,U2) - 2fGG(u 1 ,u2)) •
The mirror symmetry of the flux tube allows one to establish the above relation (3.17) between the mirror matrices with opposite and like helicities, which can be easily verified from the diagrammatic representation of the latter. Though it is not transparent from the notations in the relation (3.17), we implied one has to interchange l1 and t2 as well. Here the scattering phases are
fGG(u1 ,U2) = ij deiu1t sinh l2t 0 L
fGG(u1 ,u2) = -J d eiu1t sinh l2t 0
f(3)(u1 ,U2) = -iJ deiu1t sinh l2t 0
f(4)(u1 ,U2) = —J de'u1t sinh l2t 0
and possess only a finite number of cuts as expected.
3.2. General solution for even u -parity
In complete analogy with the fermionic case discussed in the preceding sections, we change the basis of functions as in Eq. (2.7) and then Fourier transform their linear combination as
rG(T) = r£+ (T) + iT^M) = j dke-tkr$G(k). (3.22)
Here the function ^(k) is complex contrary to the analogous one for the fermion by virtue of a similar property of the sources on the Goldstone sheet. The flux-tube equation for the former is then rewritten in the form
(3.16)
(3.17)
KG U2 (2gt) + Y+,U2 (2gt)
1 — e—
et — 1
(3.18)
K+, U2 (2gt) + K+, U2 (2gt)
1 — e—
et — 1
(3.19)
KG U2 (2gt) y+G, U2 (2gt)
1 — e—
et — 1
(3.20)
Y-G,U2 (2gt) YG,U2 (2gt)
1 — e—
et — 1
(3.21)
(p) + / ddkkP-= — / —t—(k) J n k — p J n k — p
dk0(k2 — 1) G
(3.23)
1 ^1 — — F
p + w [t—2n—2] + i0 p — û[t—2n—2] — i 0 p + M [t—2n] + i0
— u[t—2n] — i0
with the traditional convention used for the shifted (and rescaled) rapidity variable
[±t] =
= u ± ■
(3.24)
The solution for the interior region reads
(p)||p|<i = $G(P) -^(i-p
dk fk — 1\1/4 0(k2 — 1) G
n \k + 1
k— p
$G (k), (3.25)
where2
0G(p) = _L £ j e—in/4S (p — «[t—2n—2]) + ein/4S (p + Û[t—2n—2])
2 2 n=0
— e'n/4S (p — Ût—2n) — e—in/4S (p + Û[t—2n])
1/1 + p
n \ 1 — p
e—in/4 _
p / 1 + [t—2n—2] \1/4
in/4 _
p + £ [£—2n—2] y 1 — ¿J [€—2n—2]
1- 2n—2] \1/4'
— jJ [t—2n—2] I 1 + jJ [t—2n—2]
1/1 + p
e—in/4
n \ 1 — p
V ( 1 — J V—2n] \1/4
P / 1 + £ [t—2n \1/4
p + J [e—2n]y 1 — J [t—2n] J
p — « n,A 1 + « [t—2n],
For the exterior domain, we have as before the series representation
(P)l|p|>i = 0(p - 1) £cG,+(", g)e-4nng(p-1) «>i
+ 6(-p - 1) £cG,-(n, g)e-4nng(-p-1).
(3.26)
(3.27)
A formula for the partition of the product of principal value poles becomes handy here,
x — ax — b a — b
x — a x — b
+ n2S(a — b)5(x — a).
Fourier transforming back to r, we deduce
fjt = xG(t) + £ l^n—fr [-iTVo(-ir)U-(4ngn) + 4ngnV1(-ir)U-(4ngn)]
+ £ n + ix [-irVo(-ir)U+(4ngn) + 4ngnVl(-ix)U0+(4ngn)},
(3.28)
xG(T) = j dk e-'Tk$G(k) -l
1 £-i i
1 £ ^V2cos (rU[i-2n-2] + 4) - 2V2cos (tu[Z-2n] - f )
( 1 + ,}^-2n-2] X1/4 ,
^ (-iT. - [€-2n-2]) (-it,u[Z-2n-2])
/14- U[^-2nK 1/4 / x
- ein/M 1 [, „) W {-it, 2n"2]
_ pin/4
V1 - u[z-2"-'-
1 - u [Z-2n-2]
1 + . u[Z-2n-2]
1 + u [Z-2"] x1
1 - u [z-2"] )
(1 -u [1-2"]"
3.3. General solution for odd u -parity
The flux-tube equations for the Fourier transform of rj^t),
rG(t) = fuG+ (t) + irG-(T) = J dke-'kt$G(k) (3.29)
is again put in the form of a singular integral equation l
dk P ~G f dk0(k2 - 1) ~G
^ + / vr-p*G(k) = -/
u J 1 f _ J u VVZ I j u
n k - p J n k - p
®G(k) (3.30)
p + uv-2"-2] + io p -uv-2"-2]- io
+ . . +
p + u Z-2n + io p - u Z-2n - i0
whose solution is
1/4 7 Jl 1 \ 1/4
~r ~r 1 /1 + p\ 1/4 f dk (k — 1\1/4 0(k2 — 1)~G ^G(p)hpi<1=^—^(j—f) j AiTi) -i—T*G(k), (3.31)
G (p) = e—'n/4s (p — U [t—2n—20 — e!n/4s (p + U [t—2n—20 (3.32)
in/45 (p — u [t—2n]) + e—in/4S (p + U [t—2n])
1 1 + p 1/4
n V1 — p
e—in/4
P ( 1 +U [t—2n—2] \1/4
p + u V—2n—2] ^ 1 — u [t—2n—2]
-p / ^ U[t—2n— 2] \1/4
_ ein/4 1 t 1 U \
p — U V—2n—1 + U [t—2n—2]<
P / 1 + U [t—2n] V/4
1 1 + p 1/4
n V 1 — p
n +U [t—2n]y p + U [t—2n^ 1 — U [t—2n] )
— e—in/4 P / 1 — U[t—2n] V/4'
p — u [t—2n]\ 1 + u [t—2n] y
and the outside function is again determined by the series (3.27), where one obviously dresses all coefficients with tildes. Fourier transforming it back (3.29), we deduce
PGM = xUG(T) + £ 4Ggn(—f! [—irVo(—iT)U-(4ngn) + 4ngnV1(—ir)U-(4ngn)] + £ 4G n( + gT I—!TVo(—i'T)Ui+(4ngn) + 4ngnVx(—i'r)U+(4ngn)],
(3.33)
xG(T) = j dk e—'Tkffî(k) —1 1 t—1 i
= i E — i^V2sin (rU[t—2n—2] + 4) + i^V2sin (rU[t—2n] —
+ e—1 +U [t—2n—2 J ^ (—iT —U [t—2n—2])
/ U[t—2n—2] \1/4 )
— ein/^ 1 Urt 2 J w f—it, U[t—2n—2]) 1 +U [t—2n—2]
1 \1/4
1 -u [^-2"]
1 - u [t-2"] \1/4
j w (_iTt -u[^-2n]) \ 1/4 1
1+*-«) *(-^«-0 .
3.4. Quantization conditions and their solutions
The quantization condition for the even u -parity function rG(4nigxm) = 0,
can be solved order-by-order in the inverse 't Hooft coupling with the result
cG'±(n,g) = (8ngn)±l/4
+ ^Gt" + O(1/g2)
where the explicit a and b coefficients are found to be
(3.34)
(3.35)
aG+(„) = - 2lT(n + 1 ) /0G'+(u), u () T(n + 1)T2( 1 )X0 ()'
bG'+(") =
2lV(n + 1 ) Y(n + 1)T2( 1 )
--+ - ln2
aG,-(n) = -
x^ (u)
'-T(n + 3 ) g,-
2r(n + 1)T2( 3 )
X0 , (u),
(3.36)
(3.37)
---ln2
(X0G,+(u) - 8/1G0,+ (u)) - xG+(u)
+ 3L (3x0G,+ (u) - 32XG0 + (u))
bG,-(n) = -
• r(n + 4 )
2r(n + 1)T2( 4 )
---+ - ln2
16 + 8
G,+ / N X0 (u)
(3.38)
--+ - ln2
16 8 .
(xG (u) - 8xG (u)) + x 1 (u) + 32n (5xG,-(u) - 32xG0-(u))
respectively. Here, we introduced inhomogeneities arising in the left-hand side of the quantization condition,
xG^ni'gjXm!) _ g G,±(, I
V0(±4ng|xm j) x0 (u) 4ng
x 10 (u) , G,± / , --+ x 1 1 (u)
+ O(1/g2),
x0 (u) = T
1 /1 + u
1 ± u \ 1 — u
xG (u) = ±27!9u
1 /1 + u
1 ± u \ 1 — u
1 /1 - u\
1 ^ u \ 1 + u
1 /1 -u
1 ^ u \ 1 + u
(3.39)
(3.40)
(3.41)
XiQ (U) = —
/jo (u) = +
(1 + U )2 \1 — U 1 /1 + U ^/4
1 — U
(1 — U )2 \1 —
(1 — U )2 \1 + U 1 /1- U ^/4'
(1 + U +
The defining condition for the odd u -parity coefficients fG(4nigxm) = 0
provides the result
cG,±(n,g) = (8ngn)±1/4 with a and b being
+ O(1/gz)
(3.42)
(3.43)
(3.44)
(3.45)
aG,+(n) = —
2tT(n+ 4 ) ~g,+
bG,+(n) =
P(n + 1)T2( 4 )
2tT (n + 4 ) P(n + 1)T2( 4 )
û xg (u),
--+ - ln2
^G, — (n) = —
~G (u)
tP(n + 4 ) _G.
2r(n + 1)T2( 4 )
3i ~~0
(3.46)
(3.47)
---ln2
(xG,+(U) — 8jfG0 + (u)) — X G,+ (U) + 32n (3XX0G,+ (U) — 32X(1G0,+(U^ } ,
~G tr(n + 4 )
bG,—(n) = — ( 4 )
2T(n + 1)T2( 4 )
---+ - ln2
16 + 8
G,+ X0, (u)
(3.48)
--+ - ln2
16 + 8
(xi? (u) — 8xg; (u)) + xfi (u)
+ i5^ (u) — 32XG0 (u)) [ ,
where the functions xx (u) arise from the expansion of the source xx XuG(±4nig|xm|)
_ t ~ G,± ( ) 1
V0(±4ng|xm|)' = ~~0 (U) 4ng
X10 (U) , ~G,±, ,
--+ ~~11 (u)
+ O(1/g2),
~G,±s \ , 1
X0 , (u) = ±
1 + u \1/4
1 ± U v 1 — U J
1 — U
1 ^ U \ 1 + U
G ± n
~~11 (u) = du
~~ÎQ + (u) = +
1 ± U \ 1 — U
1 — U
1 ^ U \ 1 + U
(1 + U )2 \1 — u
1 — U
(1 — U )2 \1 + U
(3.49)
(3.50)
(3.51)
(3.52)
~G,-( ) 5 x\0 (u) = -
(1 -u )2 \1 -u
(1 + u )2 V.1 + u
(3.53)
3.5. Strong coupling expansion
Using the just determined expansion coefficients, we can deduce the 1 /g expansion of the flux-tube functions, which are
'1 1/4 1 - u N '
n,u(T) = T
2V2\1 +
W ±(T,U)
2V2 \1 -
1 + u \1/4iT ±
W±(t, -u) + iiS±,- sin(ru)
T 4ng u
1 /1 -U
272 U + U
W ± (T,U)
1 /1 + u.
—=-7 W±(t, -u) + i8++ cos(tu)
2^2 \1 -u J
^x01,-(u) in 3 \ . ^x01,+(u) (n 3
—— - + - ln2 V±(t) ± ——---ln2
4ng \ 8 4 J 1 4ng \ 8 4
x [V1±(r) t 4rV0T(r^ + O(1/g2)
rlu(T) = -
2V2V1 + u
1 + U \1/4
2V2 \1 -
W ±(T,U)
W±(t, -U) - H8±,+ cos(rU)
/ 1 + u \ i/4
272 \1 -U
J^ / X 1/4
272 \1 + u
W ± (T,U)
W±(t, -u) + iS±,- sin(rU)
¿x.1 (u) in 3 \
^^ '- + - ln2 V±(t) +-
8 4 / 1 ( ^ 4ng
x [ V±(t) t 4rV,T(r)] + O(1/g2)
¿xo,+(u) (n _ 3 ln2
(3.54)
(3.55)
for the even and odd u-parity, respectively. Here 5++ = S__= 1 and S+- = 5_+ = 0. Substituting these solutions into the scattering phases, we find
\ ¿^2 A(a), ... 1
f GG (u 1 ,u2) = AGG(h1,U2) + 4g
„(a), , . 3ln2 {a)
BGG (U1,U2) + —— cgg(U1,U2)
+ O(1/g2 )
(3.56)
where the linear dependence on t's holds only up to the order in 1 /g displayed and it becomes nonlinear beyond it. The explicit expressions are deferred to Appendix C.2.
At leading order, i.e., keeping just A's, we find the known expressions [1,33,34]
ln SGG(ui,U2) =
4g(Ui - u2)
ln s*gg(«i,«2) =
1 + u 1 1 — u 1
1 -u 1
1 + u 1
1 — u 2 1 + u 2
1 + u 2
1 — u 2
(3.57)
4g(u1 — u 2)
1/4 1/4
1 — u 1 \ /1 + u 2W
1 + u 1
1 + u 1
1 —u 2
1 —u 2
(3.58)
1 + u 2
— 2i
for the direct and mirror S-matrices, respectively. The subleading corrections were recently verified by a direct calculation in string perturbation theory3 [35].
The pentagon transitions at strong coupling are found by substituting the above result (3.56) into the expressions for pentagons derived in Appendix B.1,
Pg\g (u1|u2 ) = wgg(u1,u2)pg|g (u1|u2)
e— ¿/GG(u1>u2)+!/GG(u1>u2)— /GG(u1 ,u2)+/GG(u1,u2)
(3.59)
StX1i2 (u 1 ,u2)
X[—¿1][u1]x[—£2l[u2]
^¿1^1 ]X [î2'][Û2],
x[t1][u1]x [—¿2 ][u2]
x[—¿1][u1]x [¿2] [u2]
with wGG given by Eq. (B.6). Here we employed the following conventions for the shifted rapidities and rescaled Zhukowski gluon variable x[u] = gx[U],
x[±£][u]=X[u ± i4g].
Finally, let us quote the bound state measure to order O(1 /g2)
/¿g(U) = ¿2 exP
8g(1 — u2)
3^ + 12ln2 (1 + u 2 ) 16^g(1 — u2)
+ O(1/g3)
(3.60)
(3.61)
Let us point out, however, that in the derivation of this expression it is important to realize that the g — c and the square limit, used to obtain the measure from the pentagon, do not commute. Strong coupling comes first. The above 1/t2 arises solely from the 1/sti (u1, u2) prefactor in Eq. (3.59).
4. Fermion-gauge bound state transitions
The gauge bound state-(anti)fermion S-matrices are easily constructed along the same lines as the ones for a single gauge excitation and read
3 We would like to thank Lorenzo Bianchi for bringing these results to our attention.
u 1 — u2 — i 2 Si{(ui,u2) = -1 Sft(ui,U2)
U1 — U2 + i 2 exp (—(ui,u2) + 2ifjf (ui,U2)^
S*2f(ui,U2) =
u1 — u2 — i u i — U2 + i 2
■S*2f(ui,U2)
g2(—i)2
— exp (2f2(f3)(ui,u2) — 2f(4)(ui,u2^ .
Xf[M2](Mx - U2 + ¿2)
Since the prefactor, as a function of the gauge rapidity, is rational, we do not even need to pass to the Goldstone sheet to fuse this rational factor for the gluon-antifermion S-matrix. While the dynamical phases can be easily generalized for any t
fjf)(ui,U2) = J — e 2i/2sin(uii) o
Y— ,U2 (2gt) Y+ ,U2 (2gt) ' i — e—t et — i
f2(f2)(ui,U2) = J ddt (e—2t/2cos(uit) — Jo(2g)
K+ ,U2 (2gt) + K— ,U2 (2gt)
i — e—
et — i
f2(f3)(ui,U2) = —J ye 2t/2sin(uit) o
-It/2 (
K— ,U2 (2gt) K+ ,U2 (2gt)
i — e—
et — i
f2(f4)(ui,U2) = J ddt (e—2t/2cos(uit) — Jo(2gt)j
Y+ ,U2 (2gt) Y— ,U2 (2gt)
i — e—
et — i
after using exchange relations for the phases given in terms of gauge bound state flux-tube functions, see Eqs. (A.33), (A.34), (A.40) and (A.41) of Ref. [15].
4.1. Passing to Goldstone sheet
Let us pass to the Goldstone sheet since this is where we will perform the strong coupling
expansion. The scattering matrices read4
U i — U2 — i 2 Sof(ui,U2) =-—j Sc|(ui,u2)
U i — U2 + i 2 exp (—2i'f(f(ui,U2) + 2iff (ui,U2)^
4 To derive the last line in Eq. (4.8) the following formula is useful
j dt ^sin(ui 2]t)sin(u2t) + cos(ui 2]t)cos(u2t) — cos(u2t)Jo(2gt)J = ln
xxf [u2](ui ] — U2)
U1 — U2 — i 2
S*Gf(U1,U2~) = -—2 StGf(U1 ,U2)
U1 — U2 + i 2
U1 — U2 — ¿22
U1 — U2 + i 2 with corresponding dynamical phases being
I exp ^2/—3 (U1,U2) — 2/—? (u 1 ,u2))
fGGf(u1,U2) = ^^e" _. 2
dt tt iU1t sinh
Y— ,U2 (2gt) + Y+ ,U2 (2gt)
1 — e—
et — 1
1 f dt ~2J 7
cos(U2t) K_G,U1 (2gt).
/GG[(U1,U2) = —
U1t sinh
Y| ,U2 (2gt) + Y— ,U2 (2gt)
1 — e—
et — 1
dtG — Sin(u2t) Y—,U1 (2gt),
(4.10)
/(3)(U1,U2) = — if d eiU1t sinh tt 0
Y— ,U2 (2gt) Y+,U2 (2gt)
1 — e—
et — 1
1 dt — / — si 2t
sin(u2t) Y—u1 (2gt)
(4.11)
/G4f)(u1,U2) = — J dteiU1t sinh tt
Y+ ,U2 (2gt) Y— ,U2 (2gt)
1 — e—
et — 1
1 dt G
— cos(u2t) Y+—,U1 (2gt).
(4.12)
Everywhere above it is implied that Sm[u1] > t/2. We also showed results in terms of the gauge bound state flux-tube functions. Both of these expressions will be used in the next section along with the strong expansion constructed earlier to verify their consistency.
4.2. Strong coupling expansion
Employing the strong-coupling expansion of flux-tube functions worked out earlier, we can calculate the dynamical phases of the gluon-fermion pentagons. They admit the following form
\U1,U2) =
{A—f)(U1,U2) + . 16^2g [ Gf 4g
B—f(U1, U2) + -nc—f'(U1, U2)
+ O(1/g2)
(4.13)
with explicit expressions presented in Appendix C.3. The latter are the same obtained either by using gauge bound state, i.e., Eqs. (3.54), (3.55), or small fermion, Eqs. (2.55), (2.56), flux-tube functions providing a very nice check on the formalism.
Using explicit solutions, we find immediately at strong coupling
ln Scf(ui,u2) =
4V2g(wi - U2)
1 — U i
1 + U 2
U2 + 1 U2 — 1
(4.14)
1 + U ]
_1)1/V ui—lY/4
1 — u 1/ V U2 + 1 )
ln S*Gf(U1,U2) =
4\fïg(U\ — U 2)
1 —U 1
1 + U 1
U2 + 1 U2 — 1
(4.15)
1/4 1/4
m/ (u2 — + iV5
confirming leading order results of Ref. [20]. With Eq. (4.13), we can now uncover subleading terms.
The pentagon transitions at strong coupling are computed making use of the formulas derived in Appendix B.2. For the gauge bound states transitioning into small fermion (and vice versa), the results are
^G|f(U 1U2) = (—1)
U 1 — U 2 — i -J—
t 1 2 4g
U 1 — U 2 + i
x[U2] — x[ t][U 1]
X[U2] — X [+t][Ux]
x exp ^—ifGGf>(U1,U2) + ifGGf(U\,U2) — fGGf(U1 ,U2) + /G4)(U1,U2^ ,
(4.16)
■Pf|G(U2|U1) = i( 1 )
X[U2] — X[ t][Ux]
X[U2] — X [+t][Ux] x exp (i/GGf(U1,U2) — ifG>(ul,U2) — f{Gf(U1,U2) + /Gf (u1,U2)^
(4.17)
and consist in substituting the phases from Appendix C.3 along with Taylor expanding prefactors following the conventions introduced in Eqs. (2.63) and (3.60), for rescaled small fermion and gauge Zhukowski variables, respectively.
5. Constraints from Descent Equation
Before we turn to applications, let us provide an additional layer of constraints on the form of the strong-coupling expansion for pentagons. This is offered by the Descent Equation [50,51] which was recently studied within the context of the pentagon OPE in Ref. [52].
For the fermion-fermion pentagon, one can immediately find, making use of the results derived in Sect. 2, that it verifies the condition derived in [52] when one passes to the small fermion kinematics which dominates the strong coupling limit,
f ddf (1 -Xf}Mu)e-t'[Eiiu)-1]8(pf(u})Pf\f(-u\v} = (5.1)
J xf r(g)
with the right-hand side defined by the cusp anomalous dimension that admits the following strong-coupling expansion [38-42,30,43-49]
T(g) = 2g + O(1/g).
Above, we changed from the rapidity variable u to the Zhukowski xf via u = (xf + xf 1 )/2 in the integration measure and adopted the small fermion energy and momentum dispersion relation from Ref. [9]
Ef(u) = 1 + Of
Pf(u) = ^^Xf + O(xf). 2g f
Another check involves the fermion-gluon pentagon, see Eq. (39) in Ref. [52]. Passing in that relation to the fermion and Goldstone sheets for fermions and gauge excitations, respectively, we find
f dXf (1 -XfVf(u)e-r'[Ef(u)-1]S(pf(u))y d^G(v)
Pfjo( u | v)
X +[V ] X-[V ]
2ig ~T(g)
d^G(v)
X [V ] X +[V ]
- 2(Eg(v) + ipG(v))
where we introduced a differential of the integration measure for later convenience that includes the propagating "phase" factor
d^p(v) = ^Mp(v)e-tEP(v>+iapP(v>. 2n
with p = G for the case at hand. A simple counting of powers of the 't Hooft coupling immediately exhibits the fact that this equation relates contributions at different orders in g2, i.e., its left-hand side requires effects an order higher in coupling compared to its right-hand side. Using the explicit strong coupling solutions from the previous section (for I = 1), we can expand the left-hand side in the vicinity of xf = 0,
PfjG(-ujv)
X +[v ] X -[v ]
iXf 2g
X [v] X +[v]
- 2(eg(v) + ipG(v))
reproducing the one on the right. Here the energy and momentum of a single gauge excitation are [9]
Eg(v) ~
1 + v\1/4 /1 -v ^/4' +
Pg(v) ~
1 + v 1 - v
1 - v 1 + v
at leading order, with subleading terms in coupling which can be extracted from Appendix D.2 of Ref. [9].
6. Application
As an immediate application of the just derived strong-coupling results, we consider the
X1X4 component w6xix4) of the NMHV hexagon,—a function of three conformal cross ratios t, a, 0,—in the OPE limit t ^to. Though we systematically constructed the 1/g expansion in the previous sections, we will restrict our consideration below to leading effects in g only in order to observe the emergence of the classical string area from the summation of the pentagon OPE series. The study of subleading terms is much more cumbersome and is postponed to a future study.
We start our analysis with the consideration of the contribution of the fermion, that carries the
(X1X 3)
Grassmann quantum numbers of the W6 4 component of the hexagon, along with the bound state of t gluons, i.e., the states \i(u)f(v)). Thus, we have to resum the series
W(X1X4) = £ ei(t+1/2)0 Wtf (6.1)
where the individual contributions admit the form
w _ f f dpG(u)dfv)(-i)xf[v] (62)
tf = JJ \PG\f (u\v)\2 * (.)
To make notations in the integrand more compact, here and below \Pp\p'(u\v)\2 stands for \Pp\p'(u\v)\2 = Pp\p'(u\v)Pp'\p(v\u). Above, the differential measures were introduced in Eq. (5.5) and the integration contour for the small fermion is Cf = (-to, —2g) U (2g, to). For the gluon it is bound to the interval CG = (—2g, 2g), since outside of it the gauge excitation behaves as a giant hole, i.e., its energy and momentum scale as a first power of 't Hooft coupling g, and induce only exponentially suppressed contribution to the Wilson loop. By virtue of the complementarity of the fermionic and gluonic domains, we cannot hit the pole in (4.16) at strong coupling.
Then, at leading order in strong coupling \ Pf\G \2 ~ 1 and the integral over rapidities factorizes by virtue of this property. Therefore, the sum over all values of t in Eq. (6.1) can be evaluated in a closed form,5
= e'*/2f dfv)(—i)xf[v]( 1 —j du^G(u)Li2(e—TEG(u)+iaPG(u)+i^ ,
The second term in braces is of order g and is the first term in the expansion of the exponential of the minimal area. To restore the latter, one has to resum all one-fermion-multiple gauge bound states contributions. For N of these bound states accompanying the fermion, we find
5 Here we employed the well-known series representation of the dilogarithm Li2(z) = YltTO 1 zt/t2. Let us point out
that comparing the obtained expression with Eq. (F.46) of Ref. [2], one has to realize that the parameter i in this reference
is related to the angle 0 via ¡i = — ei0 as stated after Eq. (F.51). So the argument of the dilogarithm comes with a plus
eiW2 N !
h,...,lN = 1
J2 ei^(ll+-+lN) i dfvX-if] d^G(ui)...d^G(uN)
=1 \Po\f(uj \v)\2U N>j = 1 \PG\G(Uk \ u j>
Again by virtue of the scaling PG|G ~ 1, we observe factorization and, after the summation over N, we deduce
wtax43) = ¿*/2j dfv)(-i)xf[v] exp( -J ^^gmU2 (e-TEG(u)+-PG(u)+i^ _
Let us clarify that here and in Eq. (6.3), hg stands for the single-gluon measure. Adding to this the effect of antigluon bound states, we modify the exponent by an addendum that differs from the displayed term by a mere sign change in front of 0. In this manner, we recover the gluon portion of the minimal area in the t ^^ limit of the NMHV amplitude, which obviously contains an overall factor of integrated fermionic measure that is responsible for quantum numbers of the component of the superWilson loop under study.
The contribution to MHV amplitude at strong coupling receives an extra effect from an elusive excitation of mass two [3,27,28]. As was first explained in Ref. [12] within the OPE framework, it is not an elementary but rather a virtual composite state of small fermion-antifermion pair that comes on-shell as a bound state at infinite coupling. This idea was further pursued in an effective framework of Ref. [20] that assumed the existence of bound states of these mesons to reproduce the result from Thermodynamic Bethe Ansatz [3,27,28].
For the case at hand, we thus continue with the contribution of |f(ui)f(vi)f(v2)) state to the NMHV hexagon
1 f , , , Mf(ui)^f(vi)Mf(v2)
rV(ff)f =- I duidvidv2-T
(ff)f 2!i! J i i 2 |Pf|-f(ui|vi)Pf|-f(ui|v2)Pf|f(vi|v2)|2
xf[vi]xf[v2]„
X -—-fti(ui,vi,v2), (6.6)
xf[ui]
where Ri is a matrix part of the transition. The form of the latter for any internal symmetry group quantum numbers was recently worked out in Ref. [53]. What is important for the current analysis is that it has the following generic form
pn(un, vn +i)
Rn(UN, VN+l) =
j-[(Vj - Vi)2 + 1] ni>kKut - uk)2 + 1] nm,„[(um - v„)2 + 4] '
and possesses poles expected for nonsinglet transitions [12,15]. The polynomial in the numerator is of degree 22N-1 in variables uN = (u1, ..., uN) and vN+1 = (v1, ..., vN+1). The lowest nontrivial one is
P1(u1,v1,v2) = 40 + 6u2 + 4v2 - 2v1v2 + 4v^ - 6u1(v1 + v2). (6.8)
Rescaling the fermionic rapidities with the coupling constant, uj = 2guj etc., one observes that the Wf would be suppressed compared to the contribution of gluons analyzed above. However, there is a subtlety here, pointed out in Ref. [12], that the integration contour gets pinched by the aforementioned poles as g ^œ. Thus one has to move the integration contour
[+2]M -1
for u1 to the lower half-plane picking up two poles along the way u1 = vj - 2i (j = 1, 2). The latter induce leading order effect in coupling, on the same footing as gauge fields, and read
W(ff)f = y dfV2)(-i)Xf[V2~\
Of2 [vi] |Pf|f(v^2 |V2)|2|Pf|f(v^2 |V2)|2(V1 - v2)(vpj - V2)
where we dropped subleading contributions from the deformed contour. Here the composite fermion-antifermion measure is [12]
Hf(v + i)^f(v + i)
f(v) = -|P r , -I-(6.10)
11 |Pf|f(v + i|v - i)|2
with the energy/momentum of the composite excitation being Eff(v) = Ef(v + i) + Ef(v - i)/ pff(v) = pf(v + i) + pf(v - i). Making use of the explicit expressions for the pentagons at strong coupling, one finds that the expression accompanying measures in Eq. (6.9) goes to minus one at leading order, yielding a product representation of the single fermion accompanied by the (ff)-pair propagating in the OPE channel. Generally, for N (ff)-pairs, we have
njli ^Mf(M!0n^+11 dfvj)
W(-ff)Nf = -— I 71=;--—;—;—"2
)■ )2 I K - P^iWi/i |u
n-(N - 1)l/|
№N f N !(N - 1)! j | EI j>i Pf|f(Vj lv,)2U > Pfif(ui luk)Um,n Pilf(umlvn)l2
Xf[Vi Xf[ui
X ^^[uU1^^ ( i~)Xf[VN +1 ]Rn(UN, VN+1). (6.11)
The polynomial RN obeys a very important property: taking the residue of RN, for instance, at u1 = v1 + 2i, yields
res Rn(un, vN+1 )
u 1 = V1 +2i
nj>1[Vj - V1][Vj - V1 - i]Y[k>1[uk - V1 - 2i][uk - V1 - i]
Pn-1 (uN-1, VN)
(6.12)
rij>i>i[(vj - Vi)2 + imi>k>i[(ui - uk)2 + irnm,„=i[(um - v„)2 + 4] '
with the polynomial of a lower degree. Thus, we do not need the explicit form of RN here. Consecutively taking the residues, we find
(-1)N f
W(ff)N f = dfVN+\)(-l)Xf[VN+\]
f N x [
x / tldf(Vj)xL[
X™ [Vj ] 1
' •m + 2i l Vn)
. 1 x[-2][Vj ] n lPflf(Vm + 2i l Vn)l2
j N +1>n=m
EI (Vj - Vi)(Vj - Vi - i)l Pflf(Vi l Vj) l2 N+1>j>i
n (vj - vi)(vj - vi + i)\Pf\f(vi + 2i\vj + 2i)\2
+.... (6.13)
Here the displayed expression is responsible for the exponentiation of the (ff)-pair exchange. The ellipsis stand for effect of other poles which induce terms proportional to lesser powers of the composite measure (6.10). The solution of this combinatorial problem yields contributions corresponding to scattering of fermion bound states [20] which together with single pair propagating in the OPE channel results in dilogarithm expected from Thermodynamic Bethe Ansatz [3,27,28] at leading order in strong coupling,
= j*/2J dfvK-f] exp(J gf Li2 (-e^Wf))) .
(6.14)
In a similar fashion, one can work out mixed terms with both fermionic pairs and gluon bound states. The outcome of this consideration is that the complete leading order result is given by a single exponent with the argument determined by the sum of individual contributions discussed above. A detailed consideration is deferred to a future publication.
7. Conclusions
In this paper we initiated a systematic study of the strong coupling expansion for pentagon transitions in the OPE approach to the null polygonal superWilson loop. The framework is a generalization of a previous consideration [30,31] for the cusp anomalous dimension, i.e., the vacuum energy density of the flux tube. While we addressed 1 /g perturbative series, we did not include exponentially suppressed contributions in our analysis. These can be recovered in a straightforward fashion from explicit all-order representation of the flux-tube functions for relevant excitations. Presently, we considered gauge-field bound states and fermions. Their flux-tube functions can be used to find all other pentagon transitions (to complete the list of the ones explicitly given in the main text) in the perturbative string regime by means of exchange relations except the one for the hole transitions which require a separate calculation. The contribution of the latter was not addressed here with the focus being rather on the emergence of the minimal area in NMHV amplitudes. It was argued in Ref. [19] that all multi-scalar exchanges have to be resummed and were shown to induce kinematic-independent leading order effects in addition to the area for MHV case. For NMHV case, this question was recently addressed in Ref. [54]. We demonstrated there the factorization of contributions of near-massless scalars from the helicity-dependent massive particles carrying the quantum numbers of Grassmann components in question of the superWilson loop and provided a concise formula for their resummed short-distance behavior.
Acknowledgements
This research was supported by the U.S. National Science Foundation under the grants PHY-1068286 and PHY-1403891.
Appendix A. Special functions and integrals
In the body of the paper, we introduced the following special functions. The function W is related to the hypergeometric function of two variables [55] and reads
V2 } ( 1 + k \1/4 zk P
W(z,u) = — dki--- ez--. (A.1)
n J \1 - k / k - u
While V and U are related to the confluent hypergeometric function of the second kind and admit the following integral representations [31]
V2 f ( 1 + k\1/4 ekz
V"(z) = v/ dk{ —k) ÔW ' (A2)
1 r k + 1\^1/4e-k(z-1)
U±(z) = - dk[^—\ -. (A.3)
n 2 J \k - 1) (k t 1)n 1
Depending on the sign of z, these functions develop different asymptotic behavior at z ^ ±x. Up to exponentially suppressed contributions, the power series in i/z can be constructed from the following integrals,
W(±,z|.u)|z|_x,±es^ h * - (. - ^ r
X (i----^ , (A.4)
V (i T um) ' ' ;
e|z|(2 |z|)±(i/4-n/2)-i+n/2 X V»(±|z|)||zKx^ e (2|Z|)2n-3/2;r-f d- e---T(i/4-n/2)-n/2
( - \±(i/4-n/2)-n/2
X i - — , (A.5)
V 2|z|/ ' ' ;
obtained from above by a simple transformation of the integration variable. Similarly an equivalent representation for Un (n = 0, i) reads,
x R \Ti/4
d- e---±i/Mi + f) , (A.6)
U±(z) = (2z)-(4±1)/4 / dp e-88 ±1/^1 + A) 0
1 œ ( 8 \-(2t1)/4
U±(z) = 2(2z)-(2T1)/4 J dpe-88-(2±1)/M1 + 2z) . (A.7)
Explicitly, one finds
W(±\z\,u)
V0(±\z\)
1 (4 T1) 2
± ' T )2 + O(1/z2), (A.8)
\z\^œ u T 1 4z(u T 1)2
Vi(±jzj)
Vb(±|z|)
jzj^OT
14 t 1 2
= -2(1 t l)jzj±- + ^- + O(1/z2), 2 16|z|
U* (z)|^ = (2z)-(4±1)/4 r U±(z)jz^ = (2z)-(2t1)/4 r
4 ± 1 4
1 T ■
1 ± ■
4 ± 1 327 4 ± 5
+ O(1/z2)
+ O(1/z2)
(A.1b) (A.11)
with subleading terms eagerly evaluated from Eqs. (A.4), (A.5) and (A.6), (A.7) by Taylor expanding the integrand and computing the resulting integrals using the definition of the Euler Gamma function.
In the main text, we also introduced different parity components of V and W for the imaginary value of their argument z = — it. They are W(—ir, u) = W+(t, u) — iW-(r, u),
V2 f „ (1+k \1/4 , .. P
n J \1 — k J —1
^ f dk(i±k y... P
n J \ 1 —kj —1
W+(t, u) = — dk
cos(rk)
k — u
(A.12)
W —(r,u) =
sin(rk)
k — u
and Vn(—ix) = V+(t) — iV~(r),
V2 f /1 + k\1/4 cos(rk) dk
v+(t) = dk( 1 —1
— k/ (1 + k)n' 1 + k\1/4 sin(rk)
(A.13)
V-(t) = — f dk(1- I -
n( ) W V1 — V (1 + k)n
Finally, the only two integrals that are needed to solve the singular integral equations as well to derive the explicit expressions for all dynamical phases quoted below in Appendix C are the following
¡k( i—L\1/4 _L=M i—1)'/4—V2,
n \1 + k) k — q \q + 1
dk f 1 — k\ 1/4 P n \ 1 + k ) k — p
1 — P 1 + p
— V2.
(A.14)
(A.15)
These are valid for |q | > 1 and |p| < 1, respectively, Appendix B. Gauge pentagons: to Goldstone sheet and back
In this appendix we will construct pentagons for gauge field bound states. The initial point of this consideration is transitions for a single gluon undergoing a transformation into the same
or another flux-tube excitation. On the physical sheet, the analytical properties of the flux-tube functions are quite complex due to the presence of an infinite number of cuts [—2g, 2g] that are equidistantly separated along the imaginary axis starting at Sm[«j = ±2 and going to infinity. So it becomes problematic to construct the bound state observables by fusing single-particle once staying in the kinematical region of rapidities — 2g < u < 2g. A way out of this complication is to make an analytic continuation to the Goldstone (or half-mirror) sheet which has just two cuts for the gauge field at Sm[u] = ±2 [36]. Therefore, as proposed in Ref. [14], for technical and practical reasons it is instructive to move upwards through the first cut of the gluon excitation to the half-mirror (Goldstone) sheet
= u + i 2 + i0 — u .
(this results in the change x-[u] ^ g2/x-[u] while x+[u] stays intact) and fuse elementary excitations there, keeping the imaginary part of their rapidities above the cut, i.e., Sm[u] > 2. Once t of these gluons are fused together, one can always move back to the physical sheet, now passing to it through the top cut of the bound state [—2g Zhukowsky variables obeys the following transformation rules 2
i2, 2g + i2]. This implies that the
[u] —
~l][u] '
[u] —y x[+e][u].
B.1. Bound-state-bound-state pentagons
We start with the gluon-gluon and gluon-antigluon pentagons. These are given as usual by the ratio of the direct and mirror S-matrices (see Eqs. (3.1) and (3.2) of the main text where one sets t = 1) [1,10]
í>g2g(ui|u2) = W-1(ui,u2)
with the prefactor being
Sgg(ui,u2) S*gg(ui,u2)
Pg|g(ui|u2) = Wgg (ui,u2)
Sgg (ui,u) S*gg (ui,u2)
Wgg(ui,u2) =
g (u 1 - u2)(ui - u2 - i)
x +[u l]x- [u l]x+ [u2]x- [^2]
■[ui]x+[u2])
x-[ui]x-[u2]
x +[ui]x [u2]
x+[ui ]x+[u2] /
Going to the Goldstone sheet, we find the latter changes to
wgg(uI,u2) =
u1 - u2
x+[ui]x [u2]
x [ui]x+[u2]
u1 - u2 + i 1 -
x+[ui]x+[u2]
x [ui]x [u2].
and the S-matrices turn into Eqs. (3.16) and (3.17) with t = 1, respectively. Now, as explained in the preamble, the fusion is straightforward on this sheet as one is away from all the cuts and obtains
WGG(u1,u2) =n Y\wGG(ufk1—e1 — 1],u2k2 —i2 — l) (B.6)
k1=1 k2 = 1
= s*t112 (u 1 ,u2)
l1 x[t1]{u1i\x[—t2][u2^ (1 x[—t1][u1]x[t2][u2]
0 x[—t1][u1S]x[—'^fe]) 0
x[—t1][u1]x[—t2\[u2]) V x[£1][u1]x[£2l[u2]/
Here Sm[«t] > tk/2 (k = 1, 2). We obviously abused notations in the first line by calling the single gluon and bound state dressing factors by the same symbol. It will be always clear from the context what we are dealing with.
Passing to the physical sheet, but now through the top Zhukowski cut of t-bound state, i.e., x[—-k][uk] ^ g2/x[—-k][uk], we find
g2 ((u1 — u2)2 + ^( W-1 -2 (u1,u2) = ^1-2 (u1,u2)x[t1][u1]x[— t1][u1]x[t2][u2]x[—t2][u2] (B.7)
M g2 ^ —1
x[t1] [u 1]x[—-2] [u2]
xl 1 _ .. ,.«2„.. . W1 — g2
x[—-1] [u 1]x [-2][u2^7 V x[—-1][u1]x[—-2 ][u2] 2 x —1
x(1- g
x[-1][u1]x[-2][u2]/
which determines the gauge stack-(anti)stack pentagons when it is accompanied by the ratio of bound-state S-matrices (3.1) and (3.2)
2 —1 S-1-2 (u1,u2)
Pt t (ulju2) = W— e (u 1 ,u2)- ,
-1|t2 ( 1 2) -1-2 ( 1, 2 )st-1-2 (u1 ,U2),
2 St1-;2 (u1,u2)
P^\e, (u1|u2) = w-1-2 (u1,u2)~---- . (B.8)
-1|t2 S*-1t2 (u1, u2)
To compare with known results, let us give them in the explicit form. Using the relation
exp (2ia-1-2 (u1,u2) — 2a-1-2 (u1 ,u2^ (B.9)
r2 (1 + ^ + iu1 — iu2) x[-1][u1]x[—-1][u1]x[-2][u2]x[—(2][u2] " r2 (1 + -1 + iu1) r2 (1 + f — u) g2 ((u1 — u2)2 +
2/ (d—l) (Jb(2gt) — 1K/b(2gt) + 1 — e—!u1i—-1t/2 — eiu2t —-2'/2) b
we can cast the helicity-violating pentagon in the form
r (1 + ^ + iu1 — iu2) / g2 \ ^(u1|u2) = r (1 + 2 + u( r (1 + ^u;( (' - ^J (B-1b)
„L-S Y"2V1'2/1- g2 \—1/2
x -1]r[-2H r x[—-1I[—-2H I x[-1]r[-2] x1 x2 x1 x2 x1 x2
f t{f- 1) {J0{2gt) - 1} (joCst) + 1 - e-iU1t~l1t/2 - eiU2t~l2t^ 0
x exp (-ifz1l2 (u 1,U2) + iftHl2 (u 1,U2) + ft (u 1,u2) - f^ (u 1 ,U2 )) .
Here the dynamical phases are given in the text in Eqs. (3.6), (3.7), (3.9) and (3.10). While making use of the relation
s*i2h (u2,u)
stjt2 (u\,u2) =-----(B.11)
s,t!t2 (u1,u2)
we can take the square of the right-hand side of Eq. (B.8) to find
(-1/2r - iui + iu2) Ht1-rL + iui - iu2)
Pt1t2 (u1|u2) = ^-1--1-7\~(-—-. \ ) (B.12)
r (1 + t1 + iui r 1 + t2 - i u2 r 1 + ^ - i u1 + iu2
X (x[t1]x2-t2] - g2)1/2 (xi-t1]xf] - g2)1/2 (x[-t1]x2-t2] - g2)1/2
(x[t1]x2t2]- g2)I
X exp y D JoVg) - D (jo(2gt) + 1 - e~lUlt~ht'2 _ e!U2t-€2t/2)
x exp (12 (U1 ,U2) + f(22 (u 1 ,U2) + f(3) (u 1,U2) _ f(4) (u i ,U2 )) . Both of these expressions agree with Ref. [14].
B.2. Bound-state-fermion pentagons
Next we turn to the gauge bound-state-(anti)fermion pentagons. These are constructed from the single gauge field-(anti)fermion transitions which read [15]
2 Sgf(U1,U2) 2 -1 Sgf(U1,U2)
Pg|f(U1|U2) = Wgf(U1,U2)- , P ,-c(U1U2) = w_ (U1,U2)- ,
g|f( 2) g 11 S*gf(u1 ,U2)' g|f( 11 2) gf ( 11 2) S* gf (U 1,U2) '
(B.13)
Xf[«2] Л Xf[U2 ] \ 1 /. Xf[M2p 1
Wgf(«1,«2) = (И1 - U2 + 2 ) ,r """" I 1 - (1 - ) , (B.14)
2 x + [u1]x-[u1] \ x+ [u1]/ \ x-[ui]/
and scattering matrices quoted in the body of the paper in Eqs. (4.1) and (4.1) for 2 = 1. Going to the Goldstone sheet, we find
u1 - u2 + 2x-[u1]x[u2]-g2
WGf(U1,U2) =--2 -2. (B.15)
u1 - u2 - 2 x+[u1]x [u2] - g2
The fusion of the w factor produces
Л [2k-l-1] л ( 1Лг (u1 - u2 + i I) (x[-€][u1]x[u2] - g2)
wGf(u 1 ,u2) = \ [ wGf(u1 J, u2) = (-1) --. I w r+,1r -, r -,-2Г .
k=\ (u1 - u2 - i 2) (x[+г1[Ul]x[U2] -g2)
(B.16)
Again, we abused the notation here by calling the bound state and single gauge field prefactors by the same letter. Going back to the physical sheet, we find
xf[u2] A xf[u2] X —1
fA-1 f Xf[U2] i
Wff (U1,U2) = (-1) + (ui - U2 + i 2) rf1r -, r f1r -, 1 -
2 x[fJ[u1]x[-fJ[u1] \
2 x [-][u1]x[—-][u1] \ x[t] [u 1]
x 0—Mi)1. (B.17)
Analogously, for the gauge bound state-antifermion case, we get
w-f (u1,u2) = W—1 (u1,u2). (B.18)
In this manner we derive the stack-(anti)fermion pentagons
r>2 / I N / ,S-f(u1,u2) 2 _1 S-f(u1,u2)
P-lf(u1 |u2) = W-f(u1 ,u2)-—---, Pnf(u 1 |u2) = wtf (u1,u2)-—---,
S*tf(u1,u2) -|1 S*-f(u1,u2)
(B.19)
which read, respectively,
i (u — v + i-)xf[u2]
P-\f(u1 ju2) = - --i---TT2 (B.2b)
g (x[+t][u1l— xffo])1/2([x[—-][u1]—xffo])1/2
x exp ( —if-^1}(u1,u2) + iff)(u1,u2) + ff(u1,u2) — f(4)(u1,u2)) ,
P-|f(u1 |u2) = ig(x [+t][u1] — xf[u2])1/2([x[—-][u1]—xf[u2])1/2 (B.21)
x exp (~ifj;p(u1,u2) + ift(f2)(u1,u2) + ff(u1,u2) — f(4)(u1,u2)) ,
with dynamical phases quoted in Eqs. (4.3)-(4.6). These expressions are in agreement with Ref. [18] up to a different choice of normalization conventions.
In the main text, we also use the pentagon with flipped flux-tube excitations, i.e., Pf|t, and continued to the Goldstone sheet, Pf|G. This transition can be obtained in two steps, first, one uses the fact that on the physical sheet,
Pf|t(u2 |u1) = P-|f(—u11 —u2). (B.22)
Then use the following obvious properties of dynamical phases
fpp'2)(—u1, —u2) = —f!p2)(u1,u2), fpp/4)(—u1, — u2) = +fp(p3,4)(u1,u2), (B.23)
and only after that continuing the gauge bound state to the Goldstone sheet. In this fashion, we find Eq. (4.17).
Appendix C. Dynamical phases
In this appendix we summarize dynamical phases for fermion-fermion, gluon bound-state-bound state and fermion-gluon bound state transitions to the first nontrivial oder in 1/g. In a similar fashion, one can find the rest of transitions by means of the exchange relations, except for the hole-hole case, which requires a separate study.
C.1. Fermion-fermion case
For the fermion-fermion phases, the leading contributions are
A<f,)(Ui, Ü2) = -
/Ui - 1\1/4/U2 + n1/4 + /ui±1\1/4(«2 - n - 2 «1 — u2 I V«1 + 1/ \«2 - v V«1 - V V«2 + 1/
■/Ur-i\1/4(U2—1 \1/4 («1 + 1 \1/4(Û1+1 \1/4
«1 + U 2
u1 + 1
«2 + 1
«1 - 1
(3) -v
Af )(«1, U2) =
«1 - U 2
(X 1/4 /«2 + 1 \1/4 - («1 + 1 \1/4 («2-1 \1/4'
V«1 + 1) V«2 - 1) \Û1 - 1/ \Û2 + 1/
«1 + « 2 \ «1 + 1
1/4 1/4
«1 - 1 «2 - 1
«2 + 1
«1 + n1/V «2 + n1/4'
«1 - 1
«2 - 1
Af (U 1, «2) = -
«1 - « 2
( 51_1 X 1/4 ( X 1/4 - ( «1 + 1X 1/4 ( X 1/4
V«1 + 1/ V«2 - 1) \«1 - 1) \«2 + 1/
«1 + « 2 \ « 1 + 1
X 1/4 / X 1/4
«1 - 1 «2 - 1
«2 + 1
«1 +1 «1 - 1
«2 + 1 «2 - 1
while the subleading coefficients read
B(f^)(Ù1, «2) = -
(«1 - « 2)2 1
(«1 + « 2)2
( «1 - 1 Y/4 ( «2 + 1 \1/4 _ ( «1 + 1 \1/4 ( «2 - 1
\«1 + 1y - 1/ V«1 - 1/ V«2 + 1
( «1 - 1 V/4 ( «2 - 1 V/4 - ( «1 + 1 y/4 ( «2 + 1 «1 + 1 «2 + 1 «1 - 1 «2 - 1
2 « 2 « 22
4(1 - « 1 )(1 - «2) «1 - « 2
/«1 - 1\ 1/4 /«2 + n 1/4 + /«1 + 1\ 1/Y«2 - 1
\Û1 + 1) \«2 - 1/ V«1 - 1/ V«2 + 1
2 « 22 « 22
4(1 - « 1 )(1 - «2) «1 + « 2
/ «1 -1 V«1 +1
«2 - 1\ 1/4 /«1 + 1\ 1/Y«2 + 1
«2 + 1
«1 - 1
«2 - 1
B(3)(«1, «2) =
(«1 - «2)2 «1 - 1 1/4 «2 + 1 1/4
«1 + 1
< Ùt-1
«2 - 1 «2 + 1
(U, + U 2)2
( Ul—l4,14 ( Ui—l4,14 +
\u, +1) VU2 +1)
2 u 22 u 22
4(1 - U,)(1 - U2) U, - U2
-14,14 ( U2+1 V|4
u, + 1
2 U 22 U 2
4(1 - U,)(1 - U2) U, + U2
( Ul—14,14 ( 14,14 \U, + 1) VU2 +1/
B(f4)(U,, U2) = -
(U, - U2)2
( Ul—14,14 ( U±14,14 + \U, + l) \U2 -1
(U, + U 2)2
( Ul—14,14 ( 14,14 + \U, + 1) VU2 +1)
2 U 22 U 22
4(1 - U,)(1 - U2) u, - u2
- 14,14 (U2 + 14,14
U, + 1
2 u 22 u 22
4(1 - U,)(1 - U2) U, + u2
C(f,)(U,, U2) =
U, — 1
u, + 1
1 + U1U2
U2 - 1 U2 + 1
(l -u 2 )(l -u 2)
( Ul—l4,14 ( йi±14
\U, + 1/ VU2 -1/
U, + lV|4/ U2 + lV|4 — 2 U, - l) \U2 -1J
Ui+l^í U2 - lV/4'
U, - l) \U2 +1J
u, +1 y|4 ( ûi+l V|4' U, - l) \ U2 -1J
u, +1 V|4 ( ui—l4,14 — 2
U, - l) \U2 +1J
u, + lV/4 / U2 +1\
U, - 1) \U2 -1 /
Ul+lY^f U2 - lV|4'
U, - l) \U2 +1J
u, +1 y|4 ( Ui+l y4'
U1 - l) \ U2 -1J
U2 - lV|4" u, -1) \ U2 +1J
1 — U1U2
(1 — U 2x1 _ U 2)
! \ 1/4 / ~ 1 \ 1/4 U1 — 1 \ / U2 — 1 w
cf3)(U1, U2) = —
1 + U1U2
U2 + 1
1 —U 1U2
(1 — u 2x1 — u 2)
cf4)(U1, U2) =
/ U1 — 1 \ U1 + 1 1 + U 1U2
(1 — u 1)(1 — u 2)
( Û1—1 y4 \Û1 + 1/
1 —U 1U2
(1— U 21)(1 —U 22)
( U1—-X X 1/4
V U1 + 1
(1 — u 1)(1 — u 2)
( Û1—1 V/4 ( U_1±1 V/4
vÛ +1) VU2 — 1/
U-^Y'4+
U2 +1/
-i)"4 +
U2 — 1
U 2 + 1/
«1 + 1 \1/4 ( «2 + 1 \1/4' «1 — 1) \U2 — 1 J
«1 + n1/V«2 — n1/4~
«1 — 1) V«2 + 1 )
ùx + n1/V«2 + 1V/4' «1 — 1) \U2 — 1 J
Ù1 + n1/V«2 — n1/4' «1 — 1/ \ «2 + 1 J
«1 + 1 \1/4 ( «2 + 1 \1/4' «1 — 1/ \«2 — 1 J
C.2. Gluon-gluon case
For the gauge-gauge case, the 1 /g contribution to phases are
« 1, «2) =
«1 — M 2
+ 2niS( « 1 + « 2)
«1 — M 2
«1 + « 2
>1 — « A1/Vi +ù » ^/4 1+ 1 1— 2
1— 1
1— 1 1 + ù 1
1— 1
(3) ~ ~ 2i P
AGG( M1, u2) = —~----2nS(u 1 + « 2)
«1 + u 2
«1 — U 2
/1 — u 71/4
U + « 1
1+2y4 /1 + u 1y4/1 — »2\
1— 2 1— 1 1+ 2
1 + u 1 \ / 1 + u 2
«1 + u 2 V 1 + u 1
1/4 1/4
1— 1 1— 2
1— 1
1 — u 2\1/4' 1 + u 2)
1— 2
1 — U 2 \1/4 /1 + U 1 \1/4 /1 + U 2 \
1 — u 2/
(C.10)
(C.11)
aGG(uI, U2) = -
2i P Ui + U 2
Ui — U 2
— 2nS(Ui + u 2) 1 — U1 \ /1 + U2 ^ 1/4
ui + u 2
1 + U 1
1 — U 1 1 + U 1
1 —U 2
1 — U 2 1 + U 2
1 + U1 \ / 1 — U 2^ 1/4
1 —U 1
1 + U 1
1 —U 1
1 + U 2
1 + U 2 1 —U 2
(C.12)
and 1 /g2 corrections take the form
bGGu, U2) =
[(U1 — U 2)2] +
( 1 —U 1V/4 ( 1 + U 2 y/4 — ( 1 + U j
\1 + U 1/ \1 — U2/ \1 — U 1,
1 — U 2\1/4'
1 + U 2/
[(U 1 + U 2)2] +
/1 — uA1/4/1 — U2\ 1/4 / 1+«A1/4V 1 + 1/4 + 2i
V1 + U1) \1 +U2) V1 — U1) \1 —U2/ 1
2 U 22 U 22
4(1 — U 1)(1 — U2) U1 — u2
^—u ^ 1/4+(1/Y1—1/4
V1 + u 1
2 u 2 U 22
1 — u 2/
4(1 — U 1)(1 — U2) U1 + U2
/ 1 — U 1 \1 + U 1
— 2nS' (U1 + U 2),
1—y4+
1 + U 2
1 —U 1
/1 + U 1 \1 — U 1
1 +U 2
1 + U 2 \ 1 — U 2)
(C.13)
bjgu, U2) = —
[(U1 — U 2)2] +
[(U 1 + U 2)2] +
/ 1 — U 1 \1 + U 1
2 U 2 U 22
1 +U 2
/1 +U A1/4/1 + U 2 \
\1 — U 1/ \1 — U2/
4(1 — u 1)(1 — U 2) U1 — U2
(\1/4 (1+U2\1/4 — (1/4 \1 + U 1/ \1 —U2/ \1 — U 1
/1 — U A 1/Y1 + U 2 \ 1/4 /1 + U 1\1/4(1 —U2\ — 2
\1 + U 1) \1 —U2) V 1 — U1) \1 + U 2/
1 — U A 1/4
1 + U 2)
2 it 2 It ^
4(1 — ù 1_)( 1 — ù2) Ml + û2
/т — ù л1/Ут — û 2 \1/4 л + ù л1/Ут + и 2 \1/4' Vi + äj \i + it 2 / V i — ùj U — «2/
— 2ni Ss(ít i + ù 2),
(C.14)
BG4G(û 1, Û2) =
[(( — Ù 2)2] + / T—T ч 1/4 V_ + ù 1
[( ù1 + ù 2)2]+
~(т±Г(
2 it ^ it 2
i + Ù 2 1— 2
1 — Ù 2 1± 2
4(1 — ùi_)(i — ù2) «1 — m 2
1 — Ù 1 \ 1/4 / 1 + ù 2 4 1/4
2 it 22 it 2
1— 2 V
4(1 — ù_)(1 — Ù2) mI + M 2
1— 1 1 + M 1
1 — M 2 1± 2
+ 2ni S'(u 1 + ù 2),
i + M 1 y/4 z T—2 y/4 — 2
i — Ù 1/ V_ + " 2/
i + ù A1/Vi + m 2 \1/4' i — Ù 1/ V_ — " 2/
i + M A1/Vi — M 2 \ 1/4' 1 — Ù 1/ V_ + ù 2)
i + M 1 y/4 / _±2 y/4'
1 — Ù 1/ Vi — Ù 2/
(C.15)
cG_G( M1,112) =
1± 1 2
(i — ù ^)(i — ù 2)
1 — Ù 1 \ 1/4 / 1 + и 2 ^ 1/4
1— 1 2
(i — ù ^)(i — M
1— 1 1± 1
1— 2
1 — M 2 1± 2
C££(M 1, M2) = —
1± 1 2
(i — ù ^)(i — ù 22 )
/1 — ù A1/Vi + Й2У Vi + Ûl) V_ — «2/
i + ù 1 ч_/4 /T—2 )1/4'
1 — Ù1) V_ + Ù 2/
i + Ù Л1/4/1 + Ù 2^1/4'
1 — ù 1/ \1 — Ù 2
i + Ù 1 y/4 Í T—Ù y/4'
i — Ù 1/ \i + " 2/
(C.16)
1 — U 1U2
(1 -u 2x1 -U 2) 1- U \1/4
cgg^, U2) =
1 + U 1 1 + U 1U2
(1-u ^)(1 -U 2) 1-U \1/4
1 +U 1 1 -U 1U2
(1 -u 2)(1 -U 2) 1 -U -\1/4
1 +U 1
1 +U 2
i-)"4+
1 -U 2
1 + u 2 J
1 + U 1
1 -U 1
1 + U 1 1 U1
1 + U 1 1 U1
1 + U 2\1/4
1 - u 2)
1-U2 \1/4
1 + u 2/
1+U2 \1/4
1 - U2)
(C.17)
(C.18)
We used above the Hadamard regularization which is also known (to physicists) as the so-called +-prescription. We verified that fGG(u with the string calculation of Ref. [35].
+-prescription. We verified that f('(l(u1, u2) = /qG,(u2, u1). The above expressions coincide
C.3. Fermion-gluon case
Finally, we quote the gauge-fermion phases. These are
Agfa, U2) = --
U1 -U 2
1/4 1/4
1 -U 1 U2 + 1
1 + U1
U2 - 1
1 ~ X 1/4 / - 1 \ 1/4 1 +U 1 U2 - 1
1 -U 1
U2 + 1
U1 + U 2
1/4 1/4
1 -U 1 U2 - 1
1 +U 1
U2 + 1
1 + u A1/V U2 + n1/4
1 -U 1
U2 - 1
(C.19)
(2) -v
Agf)(u1, U2) =
U1 - U 2
1 -u 1\1/4f U2 +A1/4 +( 1 +U 1\1/4f U2 - A 1/4
1 +U 1
1 -U 1
U2 + 1
U1 + U 2
1 -U 1 1 +U 1
U2 - 1
U2 + 1
1 +u A1/V U2 + ^1/4
U2 - 1
(C.20)
Aßf (U 1, U2) =
U, — U 2
1 — U1 \ 1/4 / U2 + 1
1 + U 1
U2 - 1
1 + U, \1/4 / U2 — 1 \
u, + u 2
1 - ч 1/4 / - 1 \ ,|4 1 — u M i U2 — 1w
1 + U 1
U2 + 1
AÍ(U 1, U2) = --
u, — u 2
1 + U 1
u, + U 2
1 — U 1 1 + U 1
U2 - 1
U2 - 1 U2 + 1
1 — U , ) \U2 + 1
1 — U 1/ VU2 +1
1 + u ,y/4 / U2 +1\
+ i a/2
1 —U ,/ \U2 — 1,
+ iVÎ
(C.21)
1 — U l\,/4/ U2 + 1 \1/4 /1 + U ,\,/4/ U2 — 1 \
1 + U ,\,/4 /U2 + 1\
1 - U,) \U2 -1 /
+ iVÎ
(C.22)
BG,f)(U,, U2) =
u, + u 2
4(1 - U 2)(1 - U
/ l—Ul41/4 z U2+1 41/4
+Uj \ U2 - 1)
1 + u , /
U, — U 2
4(1 - U 2)(1 - U
/1—1 y1/4 / ^y,/4 +
Il + UJ U2 + 1/
bG2)(u,, U2) = -
1 + U,) U1 + U 2
4(1 - U 2 )(1 - U 2)
Vl + U 1 / VU2 -1)
1 + u , /
U, — U 2
4(1 - U 2)(1 - U
Il + U,) U2 +1/
l^4/U2 - n,/4~
1 - U, / \ U2 +1 )
1 + u , \,/4 / y/4' 1 - U, / \ U2 -1J
U2 - n,/4' 1 - U, / \ U2 +1 /
1 + u , \,/4 / Ü2+1 \,/4' 1 - U1 / \ U2 -1J
Bf U1, U2) = -
U1 + U 2
4(1 - u^)(1 - U£
( 1-1 X 1/4 / ¿2+1X 1/4 - / 1+1 X 1/4 / X1/4"
\1 + U J V"2 - 1/ V 1 - U1/ V"2 + 1/
U1 - U 2
4(1 - U^)(1 - U2)
( 1-1 X 1/4 (X 1/4 - ( 1+1 X 1/4 (52 + 1 X1/4'
\1 + Uj V«2 + 1/ V 1 - "J V"2 - 1/
bG4)( u 1, ¿2) =
«1 + U 2
4(1 - u^)(1 - u
( 1+1 X 1/4 / ¿2+1 \1/4 - ( 1+1 \1/4 ( «2-1 \1/4'
\1 + U J V«2 - 1/ V 1 - U1) V«2 + 1/
«1 - U 2
4(1 - u2)(1 - u2)
(1 - U1 \ 1/4 / «2 - 1
1 + U J V"2 + 1
cfu 1, «2) =
1+ 1 2
(1 - u 2 )(1 - u 2)
( 1 - U 1 \1/4 ( «2 + 1 1 + 1 2- 1
1- 1 2
(1 - u 2x1 - u 2)
(1 - U1 \ 1/4 / «2 - 1
1 + U J V"2 + 1
cfu 1, «2) = -
1+ 1 2
(1 - u 2x1 - u 2)
( 1 - U 1 V/4 ( U2 + 1 \1 + U 1/ \«2 - 1
1- 1 2
(1 - u 2x1 - u 22)
(1 - U 1 \ ' /U2 - 1
1 + U J V"2 + 1
1 + U A1/V«2 + n1/4~
1 - U 1/ \«2 - 1 J
1 + u A1/V«2 - n1/4~
1 - U 1/ \«2 + 1 J
1 + U A1/V«2 + n1/4~
1 - U 1/ \«2 - 1 J
1 + « A1/V«2 - n1/4~ 1 - U1) \«2 + 1 /
1 + a A1/Y»2 + 1y/4~
1 - U 1/ \«2 - 1 J
CGf (u 1, u2) = —
1 + u 1u2
(1 — u 1)(1 — u 2) 1 —u ^1/4
1 +u 1 1 —u 1u2
(1 — u 2)(1 — u 2)
Cfuu u2) =
1 —u 1 1 +u 1
1 + u 1 u2
(1 — u 1)(1 — u 2) 1- u ^/4
1 +u 1 1 —u 1u2
(1 — u 1)(1 — u 2) •1 — u -\1/4
1 +u 1
u^)"4+
u2 — 1 /
u2 + 1/
u2 + 1 u2 — 1
u2 + 1)
■ u 1
■ u 1
■ u 1
■ u 1
1 \ 1/4'
u2 -u2
(C.23)
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