- Email: [email protected]
The open superstring and the non-abelian born-infeld theory
SUPPLEMENTS ELSEVIER
The Open Superstring
Nuclear Physics B (Proc. Suppl.) 127 (2004) 166-169
and the Non-abelian
Born-Infeld
www.elsevierphysics.com
Theory
F. Machadoa*, R. Medinabt “Institute de Fisica, Universidade de S%QPaulo, 05508-900 %.o Paulo, SP, Brazil bInstituto de Ciencias, Universidade Federal de Itajubb, Av. BPS 1303,37500-903 Itajubri, MG, Brazil In the recent years new interest has arised in determining the Superstring Theory corrections to the Yang-Mills lagrangian. For a long time only the O(8) terms were known, together with some partial knowledge of the O(CX’~) ones. On this contribution we report in the recent progress on this subject, carried over in the last two years, emphasizing the scattering amplitude approach and the recent calculation of the five-point amplitude.
that, as long as the field strength Fpy is kept constant, the low energy effective theory describing their interactions, at tree level, is given by the Born-Infeld lagrangian:
1. INTRODUCTION
For a long time, the fn-st Superstring Theory [l] corrections to the Einstein-Hilbert [2] and to the Yang-Mills [2,3] lagrangians have been known. In the case of the Maxwell theory corrections, moreover, the infinite cz’ series also has been known for a long time ago [4], in the constant field strength regime. The theory was found to be described by the Born-Infeld lagrangian. Recently, interest has arised in the perturbative Non-abelian version of this last result [S-lo], which has been called the Non-abelian Born-Infeld theory [ 111. In this short communication we report on the higher (Y’correction terms that have been derived in the recent two years, emphasizing the scattering amplitude approach. We end presenting new results about the gluon five-point amplitude and its full functional dependence on two tensors, namely, the previously known t(s) and a newly defined t(,,,). 2. EFFECTIVE LAGRANGIAN, INFELD THEORY AND LEVEL AMPLITUDES
BORNTREE
Using sigma model techniques, in [4] it was derived a nice non perturbative result for interacting open massless bosonic strings. It was found *fabianoQfma.if.usp.br +rmedinaQunifei.edu.br 0920-56321.S - see front matter 0 2004 Elsevier B.\! doi:10.1016/j.nuclphysbps.2OO3.12.032
C,ff = 1/(27ra’)D’z
t(rj@v + 27&F,,)
.
(1)
This calculation was done in euclidean spacetime, where D is its dimension. The result in (1) is non perturbarive in the sense that it contains the sum of the infinite QI’ contributions. The first terms of its expansion are the following:
c
- +2)2) + . . .) ,
where g = (27rc~‘)(“-~)/~, F2 = FpVFfi” and F4 = Fa~FBrFysFGa. The constant term and the total derivative terms (a’F3, d3F5, etc.) have been omitted since they do not contribute to the dynamics of the theory. In (2), the fist term is nothing else than the Maxwell lagrangian and the second one is the first string theory correction to it. In the case of the non-abelian supersymmetric theory, the space-time dimension is D = 10 and only a perturbative approach, similar to (2), is possible [12].
All rights reserved.
(2)
R Mcchado, R. Medina/Nuclear
Table 1 Expression for the amplitude A = 8 g3 aI2 +
(‘2 -
-
* k3)L2) (k2
(‘1
} +
’ k3)L7)
’ k3)K5) -
(6
(6
+(cl
-
. k3)(<4 * Icq)(<4 +(b
-(Q
. h)
+(<5
’ k4)((<2
. t)(<3
(6 (6
(6
’ c4){
. 6)((;
(b
’ c4){
’ <3)(<2
. <4)
’ kZ)(h
+
-
’ k3)K4’)
-
(Q
+
* k3)(<3
. k4)K4’
’ kl)(<3
. k2)K4
((5
+
(6 (c2
’ k2)K4
’ Ic4)(c3
(6
. h)Kl
+
((1
* <3)(<2
. &)
} +
(cl
. c4)(c2
* <3){
+
* k2)
- kZ)K5’
* h)(<3
* k3)L3
(<2
. c3){
(<5
+ (51 . k3)(<4
* h)K4
. <4){
dependent factors Ks and KS
* k2)(kl
} +
* h)K5
’ k2)(<4 (6
(6
. k2)Ll
. k3)(h
((cl
-
’ k3)L7
’ k3)(<4
} + -
* k3)LZ
’ h)(k2
((Cl
’ k3)
’ k2)K5f)
. h)(k2
(<5 . k3)(kl
* b)
(6
and for the momentum
-(6
’ k3)K5
’ k2)L7)
* k4)(<4
((G
A(1,2,3,4,5)
((<2
+ -
(cl
(<2
=
023
. a24
a12
=
;
. a23
+
+
a34
K3
2.1. cxr2 terms
a12
a34
- 2<(3) 0 ’ { Q45
The other Ki,Ki’,
((k2
. k4)L3’+
+
(<5
* k2)
((kg
. k&‘-
(Q
. kl)
. k2)L7 -
. k4)K1’ . kl)K5
-
((cl
i-
* k2)L4
+
(
(<2
(6
. k2)(<4 . k4)(<3
.
. kl)(<3
. k4)(<3
k1)K3) -
* k4)K2)
+
. kl)K3)
} +
1
of indexes (1,2,3,4,5) ) .
+
* a24 a34
2Q23
+
2034
+
O(d2)
012
+
)
1 +
(Y24)
+
o(d2).
Li and Li’ factors, appearing in A, all have a’ expansions similar to the one for K2. of the effective
lagrangian
tor,
3. RECENT
+
Jp(F2)
T2
. k3)L2-
+
C eff
+zg2a
* k2)(c4
. k3)L2
of the four gluon amplitude[l3].
-
. ]c4)K4’-
+ ((1 - k2)(<4
It is very well known [2,3] that the three and four-gluon amplitudes have expressions that can be reproduced from the following effective lagrangian: =
((k3
+~+~}+2~(3)a’{2~2r+a3~lpn5+~+~+~+
(2;t)2 -{a,,‘~3,}-f{~
+2
* k3)
’ k4)K2
* k4)(<3
(<2 . k3)(6
(&,
* k3)L4’
* k3)(<4 . k)(c3
+
. kl)
+ ( cyclic permutations
K2
167
Physics B (Pmt. Suppl.) I27 (2004) 166169
,2
(3)
Here ’ tM1"1p2"2p3"3p4"4 is a completely known ten(8) sor, given as a sum of products of r]p”‘s (see [13], for example). It also comes in the kinematic fac-
1,*C29
k 2,*< 3, k3, *C47 k)= 4
t&lf’l/12~2P3~3C4~4~~
3.1.
t&l)w2v2113Y3w4.
[
. tr(F PlW F cc2y2 F P3V3 F P4V4 ) 1.
K(&,k
kl
Pl VI
52 P2
k2 y2
C3 P3
k3 v3
k4
(‘4 IL4
(4)
V4’
RESULTS
Five-point amplitude, (;Y’~terms effective lagrangian and higher terms
of the order
In the case of the five-gluon amplitude, it has been calculated recently in [8], explicitly in the gluon polarization vectors ci and in terms of momentum dependent factors Ki, Ki’, Li, Li’ (see
168
E Machado, R. Medina/Nuclear
Physics B (Proc. Suppl.) 127 (2004) 166169
details in the mentioned reference). Its expression is given on the previous page, in Table 1, together with the first terms of the CX’expansion of two of these momentum dependent factors (K2 and KS). In this table we have used the abbreviation oij = ki . kj and we have identified the p variable of [8] with (~45. Using the expression of A(l, 2,3,4,5) in Table 1, expanded up to cubic terms in cy’, we have been able to confirm completely the cubic interaction terms of [6], 43)
=
3.2. Towards five-point
a compact amplitude
formula
for
the
In [8] it was mentioned that the momentum dependent factors Ki, Ki’, Li and Lit were not independent at all. In fact, some explicit relations among them were found (see Appendix A.2 of the same reference). We have now found the following linearly independent relations, which were not considered in
PI: K2 + (Kl
- L3 + K1’ - L3’)/2
= 0,
KI-K4+K5=0,
-16 C(3) g2 a/3.
K1’ - K4’ + KS’ = 0, K5 + K5’ + (K1 - L4 + KI’
- L4’)/2
= 0,
L:! - L3 - L4’ = 0, L2 - L3’ - L4 = 0, Kl
+
(DcL~F,:*)
Fpr
(D’“’ F@p) Fpfs
1 (5)
and discarded very similar versions of them, but not equivalent, derived before [14,5]. The lagrangian in (5) has also been confirmed in [15], [7] and [lo] by other methods. Also, in [9,10] the cJ4 terms of the effective lagrangian have been proposed. The three F5 terms appearing in (5) allow us to introduce a new tensor, t(ie), by writing them as
is antisymmetric on each pair o(li vi) > where tllo) and remains invariant under cyclic permutations of them. tclo) has an explicit expression in terms of products of r]pV’s [16] in a similar way as the tcs) tensor of (3) does [13]. The t(lo) tensor will carry the new information, characteristic of the A(l, 2,3,4,5) amplitude.
- K1’ - (Lq - L4’)
= 0.
(6)
With the complete set of relations, the whole group of momentum dependent factors can be written in terms of K2 and K3, tremendously simplifying the expression for A( 1,2,3,4,5) in Table 1. The schematic formula is given in the next page, in Table 2. Its structure is reminescent to that of the four gluon amplitude, which consists in one momentum dependent factor F(aij) (written in terms of Gamma functions) and one kinematic factor K(C, k) (given in (4) ). By now, we have an explicit expression for the kinematic term Kc2)(c, k), in terms of the t(8) tensor. We do have a lengthy expression for Kc’) (C, k), but at the moment we are lacking of a short version of it, written in terms of the gluon polarizations, their momenta and the tensors t(8) and t(lo). 4. FUTURE
PROSPECTS
We are, in the present moment, deriving an expression for K(l) (C, k) which has a tensor structure uniquely depending on the t(g) and t(lo), in a linear way [16]. With this result at hand we will finally write the whole A(l, 2,3,4,5) formula as a sum of only two terms: a t(*) dependent one and a t(lo) dependent one. The first one will contain the contribution from the infinite D2”F4 terms (n = 0, 1,2,. . . .) of the Non-abelian
B Machado, R. Medina/Nuclear
Table 2 Compact formula K(2k, k) A(1 >2 73 74 75)
for A(1,2,3,4,5)
=
written
8 g3 o’2 . { Fl(q)
169
Physics B (Proc. Suppl.) 127 (2004) 166169
in terms of only two kinematic
K’%k)
+ F&q)
K’2’([,k)
expresions K(l)(<, k) and
},
where 1 Fl(Wj)
{ ~2
= a12
K’2’(C,k)
a23
=
a34
a23
a45
(~34
~34
K2
-
(a12
a34
-
a34
~45
-
~~12 a51)
K3
1,
a51
~45
~51
{
(G
+
(kl
.k2)
K(~l,Cz;C3,kg;C4,k4;55,k5)
+
(G
* kz)
WC2,
.<2)
K(h,k
2>.C 3, k 37.C 47 k 47*C 5, k 5 ) + -
(4-2.h)
K(Cl,h
+k
2>.C 3, k 37.C 4, k 4,*C 5, k 5 )
+
h + kz;6, k3;&I, h ;6, k,) ) + ( cyclicpermutations ).
The tensor structure of K(A, a; B, b; C, c; D, d) in Kc2)(C, k), depends only on t(s) (see eq. (4)). Born-Infeld lagrangian, while the second one will contain the contribution from the corresponding D2nF5 terms (n = 0,1,2,. . . .) of it. With this in mind, we pretend to test the D2F5 and the D4F4 terms that appear in the aI4 terms of [9]. Acknowledgements This work the brazilian FAPESP.
has been partially supported by agencies CNPq, FAPEMIG and
REFERENCES 1. M.B. Green, J.H. Schwarz, and E. Witten, Superstring theory, ~01s. 1 and 2, Cambridge, Uk: Univ. Pr. 1987. Cambridge monographs on mathematical physics. 2. D.J. Gross and E. Witten, Nucl. Phys. B277 (1986) 1. 3. A.A. Tseytlin, Nucl. Phys. B276 (1986) 391. 4. ES. Fradkin and A.A. Tseytlin, Phys. Lett. B163 (1985) 123.
N. Terzi and 5. A. Refolli, A. Santambrogio, D. Zanon, Nucl. Phys. B613 (2001) 64. 6. P. Koerber and A. Sevrin, JHEP 10 (2001) 003. 7. A. Collinucci, M. De Roo and M.G.C. Eenink, JHEP 06 (2002) 024. 8. R. Medina, F. T. Brandt, F. Machado, JHEP 07 (2002) 071. 9. P. Koerber and A. Sevrin, JHEP 0210 (2002) 046. 10. D. T. Grasso, JHEP 0211 (2002) 012. 11. A.A. Tseytlin, Nucl. Phys. B501 (1997) 41. 12. A. Bilal, Nucl. Phys. B618 (2001) 21. 13. J. H. Schwarz, Phys. Rept 89, No. 3 (1982) 223. 14. Y. Kitazawa, Nucl. Phys. B289 (1987) 599. 15. P. Koerber and A. Sevrin, JHEP 09 (2001) 009. 16. F. Machado, R. Medina, work in progress.
The Open Superstring
Nuclear Physics B (Proc. Suppl.) 127 (2004) 166-169
and the Non-abelian
Born-Infeld
www.elsevierphysics.com
Theory
F. Machadoa*, R. Medinabt “Institute de Fisica, Universidade de S%QPaulo, 05508-900 %.o Paulo, SP, Brazil bInstituto de Ciencias, Universidade Federal de Itajubb, Av. BPS 1303,37500-903 Itajubri, MG, Brazil In the recent years new interest has arised in determining the Superstring Theory corrections to the Yang-Mills lagrangian. For a long time only the O(8) terms were known, together with some partial knowledge of the O(CX’~) ones. On this contribution we report in the recent progress on this subject, carried over in the last two years, emphasizing the scattering amplitude approach and the recent calculation of the five-point amplitude.
that, as long as the field strength Fpy is kept constant, the low energy effective theory describing their interactions, at tree level, is given by the Born-Infeld lagrangian:
1. INTRODUCTION
For a long time, the fn-st Superstring Theory [l] corrections to the Einstein-Hilbert [2] and to the Yang-Mills [2,3] lagrangians have been known. In the case of the Maxwell theory corrections, moreover, the infinite cz’ series also has been known for a long time ago [4], in the constant field strength regime. The theory was found to be described by the Born-Infeld lagrangian. Recently, interest has arised in the perturbative Non-abelian version of this last result [S-lo], which has been called the Non-abelian Born-Infeld theory [ 111. In this short communication we report on the higher (Y’correction terms that have been derived in the recent two years, emphasizing the scattering amplitude approach. We end presenting new results about the gluon five-point amplitude and its full functional dependence on two tensors, namely, the previously known t(s) and a newly defined t(,,,). 2. EFFECTIVE LAGRANGIAN, INFELD THEORY AND LEVEL AMPLITUDES
BORNTREE
Using sigma model techniques, in [4] it was derived a nice non perturbative result for interacting open massless bosonic strings. It was found *fabianoQfma.if.usp.br +rmedinaQunifei.edu.br 0920-56321.S - see front matter 0 2004 Elsevier B.\! doi:10.1016/j.nuclphysbps.2OO3.12.032
C,ff = 1/(27ra’)D’z
t(rj@v + 27&F,,)
.
(1)
This calculation was done in euclidean spacetime, where D is its dimension. The result in (1) is non perturbarive in the sense that it contains the sum of the infinite QI’ contributions. The first terms of its expansion are the following:
c
- +2)2) + . . .) ,
where g = (27rc~‘)(“-~)/~, F2 = FpVFfi” and F4 = Fa~FBrFysFGa. The constant term and the total derivative terms (a’F3, d3F5, etc.) have been omitted since they do not contribute to the dynamics of the theory. In (2), the fist term is nothing else than the Maxwell lagrangian and the second one is the first string theory correction to it. In the case of the non-abelian supersymmetric theory, the space-time dimension is D = 10 and only a perturbative approach, similar to (2), is possible [12].
All rights reserved.
(2)
R Mcchado, R. Medina/Nuclear
Table 1 Expression for the amplitude A = 8 g3 aI2 +
(‘2 -
-
* k3)L2) (k2
(‘1
} +
’ k3)L7)
’ k3)K5) -
(6
(6
+(cl
-
. k3)(<4 * Icq)(<4 +(b
-(Q
. h)
+(<5
’ k4)((<2
. t)(<3
(6 (6
(6
’ c4){
. 6)((;
(b
’ c4){
’ <3)(<2
. <4)
’ kZ)(h
+
-
’ k3)K4’)
-
(Q
+
* k3)(<3
. k4)K4’
’ kl)(<3
. k2)K4
((5
+
(6 (c2
’ k2)K4
’ Ic4)(c3
(6
. h)Kl
+
((1
* <3)(<2
. &)
} +
(cl
. c4)(c2
* <3){
+
* k2)
- kZ)K5’
* h)(<3
* k3)L3
(<2
. c3){
(<5
+ (51 . k3)(<4
* h)K4
. <4){
dependent factors Ks and KS
* k2)(kl
} +
* h)K5
’ k2)(<4 (6
(6
. k2)Ll
. k3)(h
((cl
-
’ k3)L7
’ k3)(<4
} + -
* k3)LZ
’ h)(k2
((Cl
’ k3)
’ k2)K5f)
. h)(k2
(<5 . k3)(kl
* b)
(6
and for the momentum
-(6
’ k3)K5
’ k2)L7)
* k4)(<4
((G
A(1,2,3,4,5)
((<2
+ -
(cl
(<2
=
023
. a24
a12
=
;
. a23
+
+
a34
K3
2.1. cxr2 terms
a12
a34
- 2<(3) 0 ’ { Q45
The other Ki,Ki’,
((k2
. k4)L3’+
+
(<5
* k2)
((kg
. k&‘-
(Q
. kl)
. k2)L7 -
. k4)K1’ . kl)K5
-
((cl
i-
* k2)L4
+
(
(<2
(6
. k2)(<4 . k4)(<3
.
. kl)(<3
. k4)(<3
k1)K3) -
* k4)K2)
+
. kl)K3)
} +
1
of indexes (1,2,3,4,5) ) .
+
* a24 a34
2Q23
+
2034
+
O(d2)
012
+
)
1 +
(Y24)
+
o(d2).
Li and Li’ factors, appearing in A, all have a’ expansions similar to the one for K2. of the effective
lagrangian
tor,
3. RECENT
+
Jp(F2)
T2
. k3)L2-
+
C eff
+zg2a
* k2)(c4
. k3)L2
of the four gluon amplitude[l3].
-
. ]c4)K4’-
+ ((1 - k2)(<4
It is very well known [2,3] that the three and four-gluon amplitudes have expressions that can be reproduced from the following effective lagrangian: =
((k3
+~+~}+2~(3)a’{2~2r+a3~lpn5+~+~+~+
(2;t)2 -{a,,‘~3,}-f{~
+2
* k3)
’ k4)K2
* k4)(<3
(<2 . k3)(6
(&,
* k3)L4’
* k3)(<4 . k)(c3
+
. kl)
+ ( cyclic permutations
K2
167
Physics B (Pmt. Suppl.) I27 (2004) 166169
,2
(3)
Here ’ tM1"1p2"2p3"3p4"4 is a completely known ten(8) sor, given as a sum of products of r]p”‘s (see [13], for example). It also comes in the kinematic fac-
1,*C29
k 2,*< 3, k3, *C47 k)= 4
t&lf’l/12~2P3~3C4~4~~
3.1.
t&l)w2v2113Y3w4.
[
. tr(F PlW F cc2y2 F P3V3 F P4V4 ) 1.
K(&,k
kl
Pl VI
52 P2
k2 y2
C3 P3
k3 v3
k4
(‘4 IL4
(4)
V4’
RESULTS
Five-point amplitude, (;Y’~terms effective lagrangian and higher terms
of the order
In the case of the five-gluon amplitude, it has been calculated recently in [8], explicitly in the gluon polarization vectors ci and in terms of momentum dependent factors Ki, Ki’, Li, Li’ (see
168
E Machado, R. Medina/Nuclear
Physics B (Proc. Suppl.) 127 (2004) 166169
details in the mentioned reference). Its expression is given on the previous page, in Table 1, together with the first terms of the CX’expansion of two of these momentum dependent factors (K2 and KS). In this table we have used the abbreviation oij = ki . kj and we have identified the p variable of [8] with (~45. Using the expression of A(l, 2,3,4,5) in Table 1, expanded up to cubic terms in cy’, we have been able to confirm completely the cubic interaction terms of [6], 43)
=
3.2. Towards five-point
a compact amplitude
formula
for
the
In [8] it was mentioned that the momentum dependent factors Ki, Ki’, Li and Lit were not independent at all. In fact, some explicit relations among them were found (see Appendix A.2 of the same reference). We have now found the following linearly independent relations, which were not considered in
PI: K2 + (Kl
- L3 + K1’ - L3’)/2
= 0,
KI-K4+K5=0,
-16 C(3) g2 a/3.
K1’ - K4’ + KS’ = 0, K5 + K5’ + (K1 - L4 + KI’
- L4’)/2
= 0,
L:! - L3 - L4’ = 0, L2 - L3’ - L4 = 0, Kl
+
(DcL~F,:*)
Fpr
(D’“’ F@p) Fpfs
1 (5)
and discarded very similar versions of them, but not equivalent, derived before [14,5]. The lagrangian in (5) has also been confirmed in [15], [7] and [lo] by other methods. Also, in [9,10] the cJ4 terms of the effective lagrangian have been proposed. The three F5 terms appearing in (5) allow us to introduce a new tensor, t(ie), by writing them as
is antisymmetric on each pair o(li vi) > where tllo) and remains invariant under cyclic permutations of them. tclo) has an explicit expression in terms of products of r]pV’s [16] in a similar way as the tcs) tensor of (3) does [13]. The t(lo) tensor will carry the new information, characteristic of the A(l, 2,3,4,5) amplitude.
- K1’ - (Lq - L4’)
= 0.
(6)
With the complete set of relations, the whole group of momentum dependent factors can be written in terms of K2 and K3, tremendously simplifying the expression for A( 1,2,3,4,5) in Table 1. The schematic formula is given in the next page, in Table 2. Its structure is reminescent to that of the four gluon amplitude, which consists in one momentum dependent factor F(aij) (written in terms of Gamma functions) and one kinematic factor K(C, k) (given in (4) ). By now, we have an explicit expression for the kinematic term Kc2)(c, k), in terms of the t(8) tensor. We do have a lengthy expression for Kc’) (C, k), but at the moment we are lacking of a short version of it, written in terms of the gluon polarizations, their momenta and the tensors t(8) and t(lo). 4. FUTURE
PROSPECTS
We are, in the present moment, deriving an expression for K(l) (C, k) which has a tensor structure uniquely depending on the t(g) and t(lo), in a linear way [16]. With this result at hand we will finally write the whole A(l, 2,3,4,5) formula as a sum of only two terms: a t(*) dependent one and a t(lo) dependent one. The first one will contain the contribution from the infinite D2”F4 terms (n = 0, 1,2,. . . .) of the Non-abelian
B Machado, R. Medina/Nuclear
Table 2 Compact formula K(2k, k) A(1 >2 73 74 75)
for A(1,2,3,4,5)
=
written
8 g3 o’2 . { Fl(q)
169
Physics B (Proc. Suppl.) 127 (2004) 166169
in terms of only two kinematic
K’%k)
+ F&q)
K’2’([,k)
expresions K(l)(<, k) and
},
where 1 Fl(Wj)
{ ~2
= a12
K’2’(C,k)
a23
=
a34
a23
a45
(~34
~34
K2
-
(a12
a34
-
a34
~45
-
~~12 a51)
K3
1,
a51
~45
~51
{
(G
+
(kl
.k2)
K(~l,Cz;C3,kg;C4,k4;55,k5)
+
(G
* kz)
WC2,
.<2)
K(h,k
2>.C 3, k 37.C 47 k 47*C 5, k 5 ) + -
(4-2.h)
K(Cl,h
+k
2>.C 3, k 37.C 4, k 4,*C 5, k 5 )
+
h + kz;6, k3;&I, h ;6, k,) ) + ( cyclicpermutations ).
The tensor structure of K(A, a; B, b; C, c; D, d) in Kc2)(C, k), depends only on t(s) (see eq. (4)). Born-Infeld lagrangian, while the second one will contain the contribution from the corresponding D2nF5 terms (n = 0,1,2,. . . .) of it. With this in mind, we pretend to test the D2F5 and the D4F4 terms that appear in the aI4 terms of [9]. Acknowledgements This work the brazilian FAPESP.
has been partially supported by agencies CNPq, FAPEMIG and
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