Pomeron in the N=4 supersymmetric gauge model at strong couplings

0.21 MB·English

by  A. V. Kotikov

Pomeron in the = 4 supersymmetric gauge model at
N
strong couplings
A. V. Kotikova, L. N. Lipatovb,c,
a Bogoliubov Laboratory of Theoretical Physics
3
Joint Institute for Nuclear Research
1
0 141980 Dubna, Russia
2

n b II. Institut fu¨r Theoretische Physik,
a
J Universit¨at Hamburg

5 Luruper Chaussee 149,

22761 Hamburg, Germany
]
h
t
- c Theoretical Physics Department,
p
e Petersburg Nuclear Physics Institute
h
Orlova Rosha, Gatchina,
[
188300, St. Petersburg, Russia
1
v
2
8
8
0
. Abstract
1
0
3
We find the BFKL Pomeron intercept at N = 4 super-symmetric gauge theory in the form
1
v: of the inverse coupling expansion j0 = 2−2λ−1/2−λ−1+1/4λ−3/2+2(1+3ζ3)λ−2+O(λ−5/2)
with the use of the AdS/CFT correspondence in terms of string energies calculated recently.
i
X
The corresponding slope γ (2) of the anomalous dimension calculated directly up to the fifth
′
r
a order of perturbation theory turns out to be in an agreement with the closed expression ob-
tained from the recent Basso results.
PACS: 12.38.Bx
1 Introduction
PomeronistheReggesingularity ofthet-channel partialwave [1] responsible fortheapprox-
imate equality of total cross-sections for high energy particle-particle and particle-antiparticle
interactions valid in an accordance with the Pomeranchuck theorem[2]. In QCD the Pomeron
is a colorless object, constructed from reggeized gluons [3].
The investigation of the high energy behavior of scattering amplitudes in the = 4 Super-
N
symmetric Yang-Mills (SYM) model [4, 5, 6] is important for our understanding of the Regge
processes in QCD. Indeed, this conformal model can be considered as a simplified version of
QCD, in which the next-to-leading order (NLO) corrections [7] to the Balitsky-Fadin-Kuraev-
Lipatov (BFKL) equation [3] are comparatively simple and numerically small. In the N = 4
SYM the equations for composite states of several reggeized gluons and for anomalous dimen-
sions of quasi-partonic operators turn out to be integrable at the leading logarithmic approx-
imation [8, 9]. Further, the eigenvalue of the BFKL kernel for this model has the remarkable
property of the maximal transcendentality [5]. This property gave a possibility to calculate the
anomalous dimensions (AD) γ of the twist-2 Wilson operatorsin one [10], two [5, 11], three [12],
four [13, 14] and five [15] loops using the QCD results [16] and the asymptotic Bethe ansatz [17]
improved with wrapping corrections [14] 1 in an agreement with the BFKL predictions [4, 5].
On the other hand, due to the AdS/CFT-correspondence [19, 20, 21], in = 4 SYM some
N
physical quantities can be also computed at large couplings. In particular, for AD of the large
spin operators Beisert, Eden and Staudacher constructed the integral equation [22] with the use
the asymptotic Bethe-ansatz. This equation reproduced the known results at small coupling
constants and is in a full agreement (see [23, 24]) with large coupling predictions [25, 26].
With the use of the BFKL equation in a diffusion approximation [3, 4, 6], strong coupling
results for AD [25] and the pomeron-graviton duality [27] the Pomeron intercept was calculated
at the leading order in the inverse coupling constant (see the Erratum[28] to the paper [12]).
2 Similar results in the = 4 SYM and QCD were obtained in Refs. [29] and [30]. The
N
Pomeron-graviton duality in the = 4 SYM gives a possibility to construct the Pomeron
N
interaction model as a generally covariant effective theory for the reggeized gravitons [31].
Below we use recent calculations [32, 33, 34, 35] of string energies to find the strong coupling
corrections to the Pomeron intercept j = 2 ∆ in next orders. We discuss also the relation
0
−
between the Pomeron intercept and the slope of the anomalous dimension at j = 2.
2 BFKL equation at small coupling constant
The eigenvalue of the BFKL equation in = 4 SYM model has the following perturbative
N
expansion [4, 5] (see also Ref. [6])
λ λ
j 1 = ω = χ(γ )+δ(γ ) , λ = g2N , (1)
− 4π2 BFKL BFKL 16π2 c
(cid:20) (cid:21)
where λ is the t’Hooft coupling constant. The quantities χ and δ are functions of the conformal
weights m and m of the principal series of unitary M¨obius group representations, but for the
1The anomalous dimensions up to four loops were calculated also with the use of the Baxter equation [18].
f
2The value of this intercept was estimated earlier in Ref.[11].
1
conformal spin n = m m = 0 they depend only on the BFKL anomalous dimension
−
m+m 1
f
γ = = +iν (2)
BFKL
2 2
f
and are presented below [4, 5]
χ(γ) = 2Ψ(1) Ψ(γ) Ψ(1 γ), (3)
− − −
′′ ′′
δ(γ) = Ψ (γ)+Ψ (1 γ)+6ζ 2ζ χ(γ) 2Φ(γ) 2Ψ(1 γ). (4)
3 2
− − − − −
Here Ψ(z) and Ψ(z), Ψ (z) are the Euler Ψ -function and its derivatives. The function
′ ′′
Φ(γ) is defined as follows
( 1)k+1
∞
Φ(γ) = 2 − β (k +1), (5)
′
k +γ
k=0
X
where
1 z +1 z
β (z) = Ψ Ψ . (6)
′ ′ ′
4" 2 − 2 #
(cid:16) (cid:17) (cid:16) (cid:17)
Due to the symmetry of ω to the substitution γ 1 γ expression (1) is an even
BFKL BFKL
→ −
function of ν
ω = ω + ∞ ( 1)mD ν2m, (7)
0 m
−
m=1
X
where
λ λ
ω = 16ln2 1+c +O(λ3), (8)
0 4π2 " 116π2#
λ δ(2m)(1/2) λ2
D = 8 22m+1 1 ζ + +O(λ3). (9)
m − 2m+14π2 (2m)! 64π4
(cid:16) (cid:17)
According to Ref. [5] we have
1 π 165
c = 11ζ 32Ls πζ 7.5912, (10)
1 3 3 2
− 2ln2 − 2 − 16 ≈ −
(cid:18) (cid:16) (cid:17) (cid:19)
where (see [36])
x y
Ls (x) = ln2 2sin dy. (11)
3
− 2
Z0 (cid:12)(cid:12) (cid:16) (cid:17)(cid:12)(cid:12)
Thus, therightmost Pomeron singularity ofthe(cid:12)partial w(cid:12)ave f (t) in theperturbationtheory
(cid:12) (cid:12) j
is situated at
λ λ
j = 1+ω = 1+16ln2 1+c +O(λ3) (12)
0 0 4π2 " 116π2#
for small values of coupling λ. In turn, the anomalous dimension γ also has the square root
singularity in this point, which means, that the convergency radius of the perturbation series
in λ for the anomalous dimension γ = γ(ω,λ) at small ω is given by the expression
π2ω ω
λ = 1 c +O ω3 .
cr 1
4ln2 − 64ln2
(cid:18) (cid:19) (cid:16) (cid:17)
2
It will be interesting to find higher order corrections to the BFKL intercept j from the inves-
0
tigation of convergency of the perturbation theory using the analytic results for the anomalous
dimensions obtained recently [5, 12, 13, 14, 15]. Note, that the BFKL singularity for positive
ω is situated at positive λ = λ . But it is expected, that with growing ω the nearest singu-
cr
larity, responsible for the perturbation theory divergency will be at negative λ. Positions of
both singularities can be found from the perturbative expansion of γ with the possible use of
appropriate resummation methods (cf. [12]).
Due to the M¨obius invariance and hermicity of the BFKL hamiltonian in N = 4 SUSY
expansion (7) is valid also at large coupling constants. In the framework of the AdS/CFT
correspondence the BFKL Pomeron is equivalent to the reggeized graviton [27]. In particular,
in the strong coupling regime λ
→ ∞
j = 2 ∆, (13)
0
−
where the leading contribution ∆ = 2/√λ was calculated in Refs. [28, 29, 30]. Below we find
next-to-leading terms in the strong coupling expansion of the Pomeron intercept. In the next
section the simple approach to the intercept estimates discussed shortly in Ref. [28] will be
reviewed.
3 AdS/CFT correspondence
Dueto the energy-momentum conservation, the universal anomalous dimension of the stress
tensor T should be zero, i.e.,
µν
γ(j = 2) = 0. (14)
It is important, that the anomalous dimension γ contributing to the DGLAP equation [37]
does not coincide with γ appearing in the BFKL equation. They are related as follows [7]
BFKL
(see also [38])
ω j
γ = γ + = +iν, (15)
BFKL
2 2
where the additional contribution ω/2 is responsible in particular for the cancelation of the
singular terms 1/γ3 obtained from the NLO corrections (1) to the eigenvalue of the BFKL
∼
kernel [7].
Using above relations one obtains
ν(j = 2) = i. (16)
As a result, from eq. (7) for the Pomeron intercept we derive the following representation for
the correction ∆ (13) to the graviton spin 2
∞
∆ = D . (17)
m
m=1
X
In the diffusion approximation, where D = 0 for m 2, one obtains from (17) the relation
m
≥
between the diffusion coefficient D and ∆ (see [28])
1
D ∆. (18)
1
≈
3
This relation was also obtained in Ref. [39].
According to (13) and (17), we have the following small-ν expansion for the eigenvalue of
the BFKL kernel
j 2 = ∞ D ( ν2)m 1 , (19)
m
− − −
mX=1 (cid:16) (cid:17)
where ν2 is related to γ according to eq. (15)
j 2
ν2 = γ . (20)
− 2 −
(cid:18) (cid:19)
On the other hand, due to the ADS/CFT correspondence the string energies E in dimen-
sionless units are related to the anomalous dimensions γ of the local operatorsas follows [19, 25]
E2 = (j +Γ)2 4, Γ = 2γ (21)
− −
and therefore we can obtain from (20) the relation between the parameter ν for the principal
series of unitary representations of the M¨obius group and the string energy E
E2
ν2 = +1 . (22)
− 4 !
Thisexpressionforν2 canbeinsertedinther.h.s. ofEq. (19)leadingtothefollowingexpression
for the Regge trajectory of the graviton in the anti-de-Sitter space
E2 m
∞
j 2 = D +1 1 . (23)
m
− " 4 ! − #
m=1
X
Note [28], that due to (22) expression (7) for the eigenvalue of the BFKL kernel in the
diffusion approximation (18)
j = j ∆ν2 = 2 ∆ ν2 +1 , (24)
0
− −
(cid:16) (cid:17)
is equivalent to the linear graviton Regge trajectory
α
j = 2+ ′t, αt = ∆E2/2, (25)
′
2
where its slope α and the Mandelstam invariant t, defined in the 10-dimensional space, equal
′
R2
α = , t = E2/R2 (26)
′
2
and R is the radius of the anti-de-Sitter space.
Now we return to the eq. (23) in general case. We assume below, that it is valid also at
large j and large λ in the region
1 j √λ, (27)
≪ ≪
where the strong coupling calculations of energies were performed [32, 35]. Comparing the l.h.s.
and r.h.s. of (23) at large j values gives us the coefficients D and ∆ (see Appendix A). 3
m
3 When this paper was almost prepared for publication, we found the article [40] containing some of our
results (see discussions in Appendix A).
4
4 Graviton Regge trajectory and Pomeron intercept
The coefficients D and D at large λ can be written as follows 4
1 2
2 2a 8a
01 10
D = 1 , D = , , (28)
1 √λ − √λ ! 2 − λ3/2
where a and a are calculated in Appendix A
01 10
1 3
a = , a = . (29)
01 10
−4 8
As a result, we find eigenvalue (23) of the BFKL kernel at large λ in the form of the nonlinear
Regge trajectory of the graviton in the anti-de-Sitter space
E2 E2 2 E2
j 2 = D +D + , E2 = 2αt. (30)
1 2 ′
− 4  4 ! 2 
 
Note, that the perturbation theory for the BFKL equation gives this trajectory at small
ω = j 1 (see eq. (1)), where
−
1 E2
γ = +iν, ν2 = +1 . (31)
BFKL
2 − 4 !
However theenergy-momentum constraint (14), leading to ω = 1 at E = 0, is not fulfilled in the
perturbation theory, because at γ 0 the right-hand side of (1) contains the pole singularities
→
which should be cancelled after an appropriate resummation of all orders.
Neglecting the term D E2/2 E2/λ3/2 at λ in comparison with a larger correction
2
∼ → ∞
a E2/λ, we obtain the graviton trajectory (30) in the form
01
2 2a E2 8a E2 2
01 10
j 2 = 1 . (32)
− √λ − √λ ! 4 − λ3/2 4 !
Solving this quadratic equation, one can derive with the same accuracy (see [32, 35])
2 E2 a +a (j 2)
01 10
= j 2 1+2 − . (33)
√λ 4 − √λ !
(cid:16) (cid:17)
On the other hand, due to (21) this relation can be written as follows
1 2 2 a01 +a10(j 2)
j 2γ = + j 2 1+2 − (34)
2√λ − √λ − √λ !
(cid:16) (cid:17) (cid:16) (cid:17)
and for j 2 >> 1/√λ we have
−
1 1
j 2γ = 2(j 2)λ1/4 1+ +a +a (j 2) . (35)
01 10
− − " j 2 − ! √λ#
q −
4 Hereweconsideronlythe calculationofthe λ−1 correctiontoPomeronintercept. Moregeneralresultsare
presented in Appendix A.
5
In particular, for j = 4 one obtains the anomalous dimension for the Konishi operator
γ = γ [32] (see also Appendix B)
K
1 1 1
2 γ = λ1/4 1+ +a +2a = λ1/4 1+ (36)
K 01 10
− " (cid:18)2 (cid:19) √λ# " √λ#
in an agreement with eq. (28).
Fortheanomalousdimensionatj 2 1/√λfrom(34)weobtainthesquarerootsingularity
− ∼
similar to that appearing at small j 1 = ω (8)
0
−
λ1/4 a
01
γ = 1+ D +j 2 D , (37)
− √2 √λ! 1 − − 1
(cid:16)q q (cid:17)
whereD (28)isequal tothecorrection∆tothegravitontrajectoryintercept withouraccuracy
1
2 1
∆ = D 1+ .
1 ≈ √λ 2√λ!
Note, that in the region j 2 < ∆, the anomalous dimesnion is complex similar to it in the
− −
perturbative regime at j 1 < ω (8). Moreover, the position of the BFKL singularity of γ at
0
−
large coupling constants can be found from the calculation of the radius of the divergency of
the perturbation theory in 1/√λ at small j 2.
−
5 Anomalous dimension near j = 2
At j = 2, the universal anomalous dimension is zero (14), but its derivative γ (2) (the slope of
′
γ) has a nonzero value in the perturbative theory
2 3 4 5
λ 1 λ 2 λ 7 λ 11 λ
γ (2) = + + +O(λ6), (38)
′
−24 2 24! − 5 24! 20 24! − 35 24!
as it follows from exact three-loop calculations [12, 28]. Two last terms were calculated by V.
Velizhanin [42] from the explicit results for γ in five loops [15].
To find the slope γ (2) at large values of the coupling constant we calculate the derivatives
′
of the l.h.s. and r.h.s. of eq. (19) written in the form
j 2m
j 2 = D γ 1 (39)
m
− m=1 "(cid:18)2 − (cid:19) − #
X
in the variable j for j = 2 using γ(2) = 0:
1 = 1 2γ (2) mD 1 2γ (2) D , (40)
′ m ′ 1
− ≡ −
(cid:16) (cid:17) mX=1 (cid:16) (cid:17)
where D is found in Appendix A in the expansion up to m = 3 (see (A.8)). So we obtain
1
explicitly
√λ
1 2γ (2) = h (λ). (41)
′ 0
− 2
6
Here
a
∞ 0i
h (λ) = 1+ (42)
0 (λ)i/2
i=1
X
is the leading contribution to the string energy in the limit j << √λ, as it is shown in (A.1)
and (A.2)).
Itisimportant, thatnowthereisanexplicit expression forh (λ)(see(A.3)). Itwasobtained
0
from the Basso result [33] by taking the value of the angular momentum J equal to two (see
an
[34]). Substituting (A.3) in (41), we have the closed form for the slope γ (2)
′
√λI (√λ)
3
γ (2) = , (43)
′
− 4 I (√λ)
2
which is in full agreement with predictions (38) of perturbation theory.
Note, that in [12] with the use of two first terms of perturbation theory we suggested the
simple quadratic equation to resum γ (2) in all orders
′
λ 1
2
= γ (2)+ (γ (2)) . (44)
′ ′
24 − 2
It turns out, that the solution of this quadratic equation indeed interpolates γ (2) between week
′
and strong coupling regimes rather well (see ref. [28]).
In particular, for large λ values, the equation (44) leads to
√λ 1
γ (2) = +1+O , (45)
′
−2√3 √λ!
whereas the strong coupling expansion of the explicit result (43) is given below
√λ 5 1 151 15 1 1
γ (2) = 1 + + +O . (46)
′
− 4 − 2√λ 8 λ 8 λ3/2 (cid:18)λ2(cid:19)!
6 Numerical analysis of the Pomeron intercept j (λ)
0
Let us obtain an unified expression for the position of the Pomeron singularity j = 1+ω
0 0
for arbitrary values of λ, using an interpolation between weak and strong coupling regimes.
It is convenient to replace ω with the new variable t as follows
0
2ω 1+t2 1
t = 0 , ω = 0 − . (47)
0 1 ω2 0 q t
− 0 0
This variable has the asymptotic behavior t λ at λ 0 and t √λ/2 at λ
0 0
∼ → ∼ → ∞
similar to the case of the cusp anomalous dimension (see, for example, [11]). So, following the
method of Refs. [11, 12, 41], we shall write a simple algebraic equation for t = t (λ) whose
0 0
solution will interpolate ω for the full λ range.
0
We choose the equation of the form
k (λ) = k (λ)t +k (λ)t2, (48)
0 1 0 2 0
7
where the following anzatz for the coefficinets k , k and k is used:
0 1 2
k (λ) = β λ+α λ2, k (λ) = β +α λ, k (λ) = γ λ 1 +β +β λ. (49)
0 0 0 1 1 1 2 2 − 2 2
Here γ , α and β (i = 0,1,2)are free parameters, which are fixed using the known asymptotics
2 i i
of ω at λ 0 and λ .
0
→ → ∞
The solution of quadratic equation (48) is given below
k2 +4k k k
t = 1 0 2 − 1. (50)
0 q 2k
2
To fix the parameters γ , α and β (i = 0,1,2), we use two known coefficients for the weak
2 i i
coupling expansion of ω :
0
ω = e˜ λ+e˜ λ2 +e˜ λ3 +... (at λ 0) (51)
0 1 2 3
→
with
4ln2 7.5912
˜
e˜ = 0.28092, e˜ = d 0.01350 (52)
1 π2 ≈ 2 − 1 16π2 ≈ −
and first four terms of its strong coupling expansion
2 t˜ t˜ t˜ t˜
1 2 3 4
ω = 1 ∆, ∆ = 1+ + + + +... (at λ ) (53)
0 − √λ √λ λ λ3/2 λ2 ! → ∞
with (see below Eq. (62)
1 1 145 9
t˜ = , t˜ = , t˜ = 1 3ζ , t˜ = 2a ζ . (54)
1 2 3 3 4 12 3
2 − 8 − − − 128 − 2
The coefficients e˜ and t˜ are unknown but we estimate them later from the interpolation.
3 4
Then, for the weak and strong coupling expansions of t one obtains
t = e λ+e λ2 +e λ3 +..., (when λ 0), (55)
0 1 2 3
→
t = √λ 1 t1 t2 t3 t4 +..., (when λ ), (56)
0 2 − √λ − λ − λ3/2 − λ2 → ∞
(cid:16) (cid:17)
where
3 11
e = e˜ , e = e˜ , e = e˜ +e˜3, t = t˜ +1 = , t = t˜ t˜2 +1 = ,
1 1 2 2 3 3 1 1 1 2 2 2 − 1 − 8
1
t = t˜ 2t˜ t˜ +t˜3 t˜ +1 = 1+12ζ ,
3 3 − 2 1 1 − 1 −4 3
(cid:16) (cid:17) (cid:16) (cid:17) 9 3
t = t˜ 2t˜ t˜ t˜2 +3t˜t˜2 t˜4 t˜ 2t˜ +1 = 2a + ζ . (57)
4 4 − 3 1 − 2 2 1 − 1 − 2 − 1 12 128 − 2 3
(cid:16) (cid:17) (cid:16) (cid:17)
Comparing the l.h.s. and the r.h.s. of Eq. (48) at λ 0 and λ , respectively, we
→ → ∞
derive the following relations
α = 4α , α = 6α , β = C α , β = C α , γ = C α ,β = C e +C e2 α (58)
2 0 2 0 1 1 0 2 2 0 2 3 0 0 1 1 3 1 0
h i
8
-1 + jo
1
0.8
0.6
0.4
0.2
z
-2 2 4 6
Figure 1: (color-online). The results for j as a function of z (λ = 10z).
0
with the free parameter α which disappears in the retionship k /k and k /k and, thus, in
0 1 2 0 2
the results (50) for t .
0
Here
C 23.99, C 5.71, C 73.65, (59)
1 2 3
≈ − ≈ − ≈
which lead to the following predictions for the coefficients e and t in (55) and (56)
3 4
6e +2C e e +4e2 +C e2 121+16(C 3C +5C )
e = 2 2 1 2 1 3 2 0.01676, t = 3 − 1 2 15.5785
3 4
− C +2C e ≈ − 128 ≈
1 3 1
(60)
and, respectively, for the corresponding terms in (51), (53) and (54)
e˜ 0.03893, t˜ 11.0192, a 8.1793. (61)
3 4 12
≈ − ≈ − ≈
Note that the results for the coefficients e , t , e˜ , t˜ and a do not depend on the free
3 4 3 4 12
parameter α .
0
On Fig. 1, we plot the pomeron intercept j as a function of the coupling constant λ. The
0
behavior of the pomeron intercept j shown in Fig.1 is similar to that found in QCD with some
0
additional assumptions (see ref. [30]).
7 Conclusion
We found the intercept of the BFKL pomeron at weak and strong coupling regimes in the
= 4 Super-symmetric Yang-Mills model.
N
9