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On the Witten vertex
Volume 180, number 1,2
PHYSICS LETTERS B
6 November 1986
ON THE WI'ITEN VERTEX D. A R N A U D O N , V. R I V A S S E A U Centre de Physique Theorique, Ecole Polytechnique, F-91128 Palaiseau Cedex, France
O. BERNIER, N. C A S T E L and D. D U H A M E L Ecole Polytechnique, F-91128 Palaiseau Cedex, France
Received 30 June 1986 The three-string vertex is computed by elementary means in the formalism of Witten. It is shown that in contrast with the standard overlap of Cremmer and Gervais, it is possible to carry out an explicitly convergent computation, without Neumann functions and multi-sheeted mappings. Numerical evidence is found that the Witten vertex for physical states reduces to the Veneziano vertex.
1. Introduction. Witten's beautiful proposal to base string field theory on noncommutative geometry [ 1 ] (at least for open strings) has attracted much interest, and some controversy has arisen on its ability to reproduce the Veneziano model [2]. In particular the relatively complicated mappings (see ref. [3] ) which are necessary to map the Riemann surfaces corresponding to Witten's interaction into the standard half-plane and to use the Neumann's functions method might have obscured the issue. In this paper we show that a completely "down to earth" approach is possible, following the suggestion in the original paper [ 1 ] to write the vertex as a product of 7r-functions. The computation is very simple and we obtain the matrix elements between arbitrary states in terms of convergent series which can be computed numerically with great precision. This allows us to check with overwhelming evidence that the on-shell couplings o f Witten's theory are indeed those of the Veneziano model, with completely elementary means. What is interesting is that the situation is in marked contrast with what occurs in the standard picture of joining and splitting o f strings [4]. There, the analytic mapping of the Riemann surface (which is flat) to the upper halfplane is simple and was obtained first [5], but to relate the corresponding Neumann coefficients to the ones obtained by writing directly n-functions and making a Fourier analysis is complicated [4] ; more important, it can be done only in a formal sense. Indeed the matrices used in ref. [4] are formal (see, e.g., the definition (2.17) in ref. [4] of the matrix B 3 ; using definitions (2.14), (2.15), note that all the coefficients are given by divergent sums). Here it is the contrary; the 6-functions are simple and only give rise to well-defined matrices, and it is the analytic mappings which are complicated. During completion o f this work we received a preprint [6] which reaches the same conclusions; a more detailed analysis o f Witten's vertex is performed and the analytic methods are developed to prove rigorously that its onshell couplings are those of the Veneziano model. 2. The c o m p u t a t i o n . The vertex proposed by Witten is
V= fDx~(o)exp[~ix270r/2)] I-I
I-I
i,U O~a~Tr/2
~[x~(ff-o)-X~+l(a)]
(1)
where by convention the index/a runs from 1 to 27, and the last value x 27 represents the bosonized ghost ~0of ref. [1]. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. 41 (North-Holland Physics Publishing Division)
Volume 180, number 1,2
PHYSICS LETTERS B
6 November 1986
We follow ref. [4] step by step. We write the modes decomposition (forgetting the i and/1 dependence when unnecessary)
x(o)=x 0 + 2
~ p~,l
(2)
Xp cospo.
We need also the Fourier decomposition of tlae half-string in terms of the modes of the full string:
2 \ +e It q~O ~ X2q+l(-1)n+q[(2n+ Xn, e -en(x2n
2 q + 1) -1 - ( 2 n - - 2 q - 1 ) - 1 1 1 ,
l
(3)
where e is +1 or - 1 depending on whether the first half (0 ~< o ~< It/2) or the second half (it/2 ~< o ~< It) of the string is considered. Clearly using (1) the modes of the third string can be written entirely in terms of the modes of strings 1 and 2, as was done in ref. [4]. But it remains to express the coincidence of the - half of string 1 with the + half of string 2. Glancing at (3), one realizes that this can be achieved by choosing all the modes of string 1 and the odd modes of string 2 as a set of independent variables. The 6-functions of (1) can then be written as expressing the other modes in terms of these independent variables. They become
l-I 6(xln+ 1 +X2znl+" +x32n 1+ )'
n~O
['I 6(x2 n - x l n +2 ~
lrq~o
n~O
(X~q+l
+X2q+l)(_l)n+q[(2n + 2q + l) - 1 - ( 2 n - 2 q - 1 ) - l ] ) ,
l~ 6(X~n-xln +2 ~ (X~q+l)(--1)n+q[(2n+ 2q+1)
n~0
~q~0
-1-(2n-2q-l)-1]).
(4)
Note that there is no equivalent of the lengths t~i of the strings or of the 13parameters of ref. [4], which is a great simplification. Of course it might be instructive to rewrite the analysis for an overlap which involves a fraction ~ of strings 1 and 2, and to see explicitly how Witten's vertex deforms into the standard one as 0 tends to 0, but simplicity is lost for a general ~. In complete parallel with ref. [4] we perform a Fourier transform to factorize the 5-function of overall momentum conservation, and to single out the zero modes. We introduce the following notations:
Un=-P2n+1
for n ~> 0,
Vn---p2n
for n > 0,
k-Po,
Anm-(2/~)(-1)n+m[(2m+ 2n+ l ) - l - ( 2 m - 2 n - l ) Fn-Ano
forn~>0,
Enm-6nm(4n)-i
Dnm-6nm[2(2n+ l)]-I
-1]
forn~>0andm>0,
forn~>0andm~>0,
forn>0andm>0.
(5)
The vertex (4) becomes, with the usual vector notation of ref. [4], ~(k 1 + k 2 + k3)6(01 +02 + v3) 6(u 2 - u I
+A(o ~ +vE)+F(k 1 +k2))5(u 3 - u 1 +Av E + Fk2).
(6)
Introducing the coherent-state basis as in ref. [4] and integrating over modes Pn for n > 0, we obtain the vertex at large times in terms of the annihilation operators an in Fock space. We define/3 n - a2n and 7n - a2n+1" The computation is tedious since it involves three successive gaussian integrations, but is straightforward (the details 42
Volume 180, number 1,2
PHYSICS LETTERS B
6 November 1986
of the computation can be found in ref. [7]). We obtain, up to a constant multiplicative factor (we rely somewhat on context for the meaning of upper indices which are either Lorentz or string indices), 27
g = ~(kl + k2 + k3) exp[~ix27 (rr/2)] exp( ~=1Q(3U, qJU,kU)) , Q([3,7,k ) = Ql(k) + Q2(fl,'),,k) + Q3(fl,7), Q1 = [-½ tFDF + 2~ tFDAG-1 tAOF] [(kl)2 + (k2)2 + (k3)2], Q2 = i,~/,k eilk ~i(2x/~/3)[DF -- (2/3)DAG-1 tADF] k / - ~ s i f f i ( 4 v r 2 / 9 ) [ EG-1 tADF] k/' Q3 = .~.(3i[(--~ij + 2)D + (~-6ii -- 4)DAG-1
tADI~/
t,l
+ ~ 7i[-50~ + (46i/- ~)EG-1E]7 / + ~ ~i(~ei/kDAG-1E)7:, i,/ i,/,k
(7)
where G - 2E + 2 tADA ' and all internal multiplications have to be done according to the ranges of definition in (5). It is proved in ref. [7] that G can be inverted through a weakly convergent Neumann series, hence that G -1 is well defined (i.e. each coefficient is given by a convergent power series). This result can be obtained by tedious elementary bounds, which we only sketch here. After some algebraic simplification one has n-1 =
Gnn
1+ 4 ~ 1 2n 3nrr2 k=O (2k + l)2
,
n~l,
Gnm
= (- 1)n+m+l
n-1 ~
4 --
3rr2(n 2 _ m 2 ) k = m
1
2k+l'
n > m >/1,
(8)
with Gnm = Gmn. Let us define A and R as the diagonal and off-diagonal parts of G. Hence G = A + R, and the Neumann series for G -1 is
G -I = ~ K hA,
(9)
n
where K - -A-1R. In ref. [7] we prove that this Neumann series is weakly convergent, hence that each coefficient is given by convergent series. Since the matrix K is not Hilbert-Schmidt, the verification is tedious and uses explicit computations at low values, and integrals and logarithms instead of discrete sums at large values [7]. It is also easy to check that all sums corresponding to internal multiplications in (7) are absolutely convergent. Therefore (7) is an unambigious definition of Witten's vertex in Fock space. Note that (7) is explicitly symmetric with respect to circular permutations of the three strings, whereas the vertex of ref. [4] had to include an (infinite) factor det B 3 to restore crossing symmetry. Hence the finiteness of the matrix elements of Witten's vertex is probably due to its more symmetric form. We can now compute the ratio of tachyon-tachyon-tachyon to vector-tachyon-tachyon coupling for onshell states. As explained in refs. [ 1,3], the ghosts decouple from this computation. Let us call P ~ the coefficient of coupling of the first mode,/31, r, of the rth string to k s. After some algebraic simplifications we obtain m-1 31 32----~--~2F1--S 4(--1)n+m [G -1] (?0 1 P01 = - p 0 1 - 3 , L n,m;~l 3•2m2(4n 2 _ 1) nm = 2 - ~
)1
"
(10) 43
Volume 180, number 1,2
PHYSICS LETTERS B
6 November 1986
Our best numerical computations (on a very small machine) involve values o f n and m up to 500, and a precision o f about 10 -5 (of course much better precision could be obtained using i.e. Pad6 techniques). The matrix G -1 is computed by the power series above. We found: 31 = 0.544336 . ~. 2X/~/3vc3 P01 . . . = 0.544331 . .
(11)
which is surely related (through a different normalization convention) to the coefficient 4/3x/~ found in ref. [2] ; hence this gives a numerical check of the method of images of ref. [2]. To obtain the ratio of t a c h y o n - t a c h y o n tachyon to t a c h y o n - t a c h y o n - v e c t o r scattering, one should however correct by a factor exp(-P00 ) since ~r(kr) 2 = 3 in the first case and F~r(kr)2= 2 in the second case. One finds P00 = - 0 . 2 6 1 6 2 8 . . . ~ log(4/3V~) = - 0 . 2 6 1 6 2 4 ....
(12)
hence the ratio is X/~/2, the correct value within the conventions of ref. [5]. (Note that our coefficients P01 are related to the coefficients N01 of refs. [4, 5] through P01 = x/--2N01. In refs. [4,5], N031 -- 1/2.) Of course our method would compute equally well any other coefficient, and equality with the "Veneziano 1'' values should hold, according to the general argument of ref. [ 1 ]. Equalities (11) and (12) are proved in ref. [6] (with again slightly different conventions; the momenta PGJ of ref. [6] are related to ours and to those of refs. [4,5], called p, by PGJ = N/2P)"
References [1] [2] [3] [4] [5] [6] [7]
44
E. Witten, Non commutative geometry and string field theory, Princeton University preprint (October 1985). S. Samuel, On the string field theory proposal of E. Witten, CERN preprint (February 1986). S.B. Giddings, The Veneziano amplitude from interacting string field theory, Princeton University preprint, January 1986. E. Cremer and J.L. Gervais, Nucl. Phys. B 76 (1974) 209. S. Mandelstam, Nucl. Phys. B 69 (1974) 77. D.J. Gross and A. Jevicki, Princeton University preprint (May 1986). O. Bernier, N. Castel and D. Duhamel, La th~orie des champs de cordes de Witten, Rapport d'option, Ecole Polytechnique, Palaiseau (June 1986).
PHYSICS LETTERS B
6 November 1986
ON THE WI'ITEN VERTEX D. A R N A U D O N , V. R I V A S S E A U Centre de Physique Theorique, Ecole Polytechnique, F-91128 Palaiseau Cedex, France
O. BERNIER, N. C A S T E L and D. D U H A M E L Ecole Polytechnique, F-91128 Palaiseau Cedex, France
Received 30 June 1986 The three-string vertex is computed by elementary means in the formalism of Witten. It is shown that in contrast with the standard overlap of Cremmer and Gervais, it is possible to carry out an explicitly convergent computation, without Neumann functions and multi-sheeted mappings. Numerical evidence is found that the Witten vertex for physical states reduces to the Veneziano vertex.
1. Introduction. Witten's beautiful proposal to base string field theory on noncommutative geometry [ 1 ] (at least for open strings) has attracted much interest, and some controversy has arisen on its ability to reproduce the Veneziano model [2]. In particular the relatively complicated mappings (see ref. [3] ) which are necessary to map the Riemann surfaces corresponding to Witten's interaction into the standard half-plane and to use the Neumann's functions method might have obscured the issue. In this paper we show that a completely "down to earth" approach is possible, following the suggestion in the original paper [ 1 ] to write the vertex as a product of 7r-functions. The computation is very simple and we obtain the matrix elements between arbitrary states in terms of convergent series which can be computed numerically with great precision. This allows us to check with overwhelming evidence that the on-shell couplings o f Witten's theory are indeed those of the Veneziano model, with completely elementary means. What is interesting is that the situation is in marked contrast with what occurs in the standard picture of joining and splitting o f strings [4]. There, the analytic mapping of the Riemann surface (which is flat) to the upper halfplane is simple and was obtained first [5], but to relate the corresponding Neumann coefficients to the ones obtained by writing directly n-functions and making a Fourier analysis is complicated [4] ; more important, it can be done only in a formal sense. Indeed the matrices used in ref. [4] are formal (see, e.g., the definition (2.17) in ref. [4] of the matrix B 3 ; using definitions (2.14), (2.15), note that all the coefficients are given by divergent sums). Here it is the contrary; the 6-functions are simple and only give rise to well-defined matrices, and it is the analytic mappings which are complicated. During completion o f this work we received a preprint [6] which reaches the same conclusions; a more detailed analysis o f Witten's vertex is performed and the analytic methods are developed to prove rigorously that its onshell couplings are those of the Veneziano model. 2. The c o m p u t a t i o n . The vertex proposed by Witten is
V= fDx~(o)exp[~ix270r/2)] I-I
I-I
i,U O~a~Tr/2
~[x~(ff-o)-X~+l(a)]
(1)
where by convention the index/a runs from 1 to 27, and the last value x 27 represents the bosonized ghost ~0of ref. [1]. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. 41 (North-Holland Physics Publishing Division)
Volume 180, number 1,2
PHYSICS LETTERS B
6 November 1986
We follow ref. [4] step by step. We write the modes decomposition (forgetting the i and/1 dependence when unnecessary)
x(o)=x 0 + 2
~ p~,l
(2)
Xp cospo.
We need also the Fourier decomposition of tlae half-string in terms of the modes of the full string:
2 \ +e It q~O ~ X2q+l(-1)n+q[(2n+ Xn, e -en(x2n
2 q + 1) -1 - ( 2 n - - 2 q - 1 ) - 1 1 1 ,
l
(3)
where e is +1 or - 1 depending on whether the first half (0 ~< o ~< It/2) or the second half (it/2 ~< o ~< It) of the string is considered. Clearly using (1) the modes of the third string can be written entirely in terms of the modes of strings 1 and 2, as was done in ref. [4]. But it remains to express the coincidence of the - half of string 1 with the + half of string 2. Glancing at (3), one realizes that this can be achieved by choosing all the modes of string 1 and the odd modes of string 2 as a set of independent variables. The 6-functions of (1) can then be written as expressing the other modes in terms of these independent variables. They become
l-I 6(xln+ 1 +X2znl+" +x32n 1+ )'
n~O
['I 6(x2 n - x l n +2 ~
lrq~o
n~O
(X~q+l
+X2q+l)(_l)n+q[(2n + 2q + l) - 1 - ( 2 n - 2 q - 1 ) - l ] ) ,
l~ 6(X~n-xln +2 ~ (X~q+l)(--1)n+q[(2n+ 2q+1)
n~0
~q~0
-1-(2n-2q-l)-1]).
(4)
Note that there is no equivalent of the lengths t~i of the strings or of the 13parameters of ref. [4], which is a great simplification. Of course it might be instructive to rewrite the analysis for an overlap which involves a fraction ~ of strings 1 and 2, and to see explicitly how Witten's vertex deforms into the standard one as 0 tends to 0, but simplicity is lost for a general ~. In complete parallel with ref. [4] we perform a Fourier transform to factorize the 5-function of overall momentum conservation, and to single out the zero modes. We introduce the following notations:
Un=-P2n+1
for n ~> 0,
Vn---p2n
for n > 0,
k-Po,
Anm-(2/~)(-1)n+m[(2m+ 2n+ l ) - l - ( 2 m - 2 n - l ) Fn-Ano
forn~>0,
Enm-6nm(4n)-i
Dnm-6nm[2(2n+ l)]-I
-1]
forn~>0andm>0,
forn~>0andm~>0,
forn>0andm>0.
(5)
The vertex (4) becomes, with the usual vector notation of ref. [4], ~(k 1 + k 2 + k3)6(01 +02 + v3) 6(u 2 - u I
+A(o ~ +vE)+F(k 1 +k2))5(u 3 - u 1 +Av E + Fk2).
(6)
Introducing the coherent-state basis as in ref. [4] and integrating over modes Pn for n > 0, we obtain the vertex at large times in terms of the annihilation operators an in Fock space. We define/3 n - a2n and 7n - a2n+1" The computation is tedious since it involves three successive gaussian integrations, but is straightforward (the details 42
Volume 180, number 1,2
PHYSICS LETTERS B
6 November 1986
of the computation can be found in ref. [7]). We obtain, up to a constant multiplicative factor (we rely somewhat on context for the meaning of upper indices which are either Lorentz or string indices), 27
g = ~(kl + k2 + k3) exp[~ix27 (rr/2)] exp( ~=1Q(3U, qJU,kU)) , Q([3,7,k ) = Ql(k) + Q2(fl,'),,k) + Q3(fl,7), Q1 = [-½ tFDF + 2~ tFDAG-1 tAOF] [(kl)2 + (k2)2 + (k3)2], Q2 = i,~/,k eilk ~i(2x/~/3)[DF -- (2/3)DAG-1 tADF] k / - ~ s i f f i ( 4 v r 2 / 9 ) [ EG-1 tADF] k/' Q3 = .~.(3i[(--~ij + 2)D + (~-6ii -- 4)DAG-1
tADI~/
t,l
+ ~ 7i[-50~ + (46i/- ~)EG-1E]7 / + ~ ~i(~ei/kDAG-1E)7:, i,/ i,/,k
(7)
where G - 2E + 2 tADA ' and all internal multiplications have to be done according to the ranges of definition in (5). It is proved in ref. [7] that G can be inverted through a weakly convergent Neumann series, hence that G -1 is well defined (i.e. each coefficient is given by a convergent power series). This result can be obtained by tedious elementary bounds, which we only sketch here. After some algebraic simplification one has n-1 =
Gnn
1+ 4 ~ 1 2n 3nrr2 k=O (2k + l)2
,
n~l,
Gnm
= (- 1)n+m+l
n-1 ~
4 --
3rr2(n 2 _ m 2 ) k = m
1
2k+l'
n > m >/1,
(8)
with Gnm = Gmn. Let us define A and R as the diagonal and off-diagonal parts of G. Hence G = A + R, and the Neumann series for G -1 is
G -I = ~ K hA,
(9)
n
where K - -A-1R. In ref. [7] we prove that this Neumann series is weakly convergent, hence that each coefficient is given by convergent series. Since the matrix K is not Hilbert-Schmidt, the verification is tedious and uses explicit computations at low values, and integrals and logarithms instead of discrete sums at large values [7]. It is also easy to check that all sums corresponding to internal multiplications in (7) are absolutely convergent. Therefore (7) is an unambigious definition of Witten's vertex in Fock space. Note that (7) is explicitly symmetric with respect to circular permutations of the three strings, whereas the vertex of ref. [4] had to include an (infinite) factor det B 3 to restore crossing symmetry. Hence the finiteness of the matrix elements of Witten's vertex is probably due to its more symmetric form. We can now compute the ratio of tachyon-tachyon-tachyon to vector-tachyon-tachyon coupling for onshell states. As explained in refs. [ 1,3], the ghosts decouple from this computation. Let us call P ~ the coefficient of coupling of the first mode,/31, r, of the rth string to k s. After some algebraic simplifications we obtain m-1 31 32----~--~2F1--S 4(--1)n+m [G -1] (?0 1 P01 = - p 0 1 - 3 , L n,m;~l 3•2m2(4n 2 _ 1) nm = 2 - ~
)1
"
(10) 43
Volume 180, number 1,2
PHYSICS LETTERS B
6 November 1986
Our best numerical computations (on a very small machine) involve values o f n and m up to 500, and a precision o f about 10 -5 (of course much better precision could be obtained using i.e. Pad6 techniques). The matrix G -1 is computed by the power series above. We found: 31 = 0.544336 . ~. 2X/~/3vc3 P01 . . . = 0.544331 . .
(11)
which is surely related (through a different normalization convention) to the coefficient 4/3x/~ found in ref. [2] ; hence this gives a numerical check of the method of images of ref. [2]. To obtain the ratio of t a c h y o n - t a c h y o n tachyon to t a c h y o n - t a c h y o n - v e c t o r scattering, one should however correct by a factor exp(-P00 ) since ~r(kr) 2 = 3 in the first case and F~r(kr)2= 2 in the second case. One finds P00 = - 0 . 2 6 1 6 2 8 . . . ~ log(4/3V~) = - 0 . 2 6 1 6 2 4 ....
(12)
hence the ratio is X/~/2, the correct value within the conventions of ref. [5]. (Note that our coefficients P01 are related to the coefficients N01 of refs. [4, 5] through P01 = x/--2N01. In refs. [4,5], N031 -- 1/2.) Of course our method would compute equally well any other coefficient, and equality with the "Veneziano 1'' values should hold, according to the general argument of ref. [ 1 ]. Equalities (11) and (12) are proved in ref. [6] (with again slightly different conventions; the momenta PGJ of ref. [6] are related to ours and to those of refs. [4,5], called p, by PGJ = N/2P)"
References [1] [2] [3] [4] [5] [6] [7]
44
E. Witten, Non commutative geometry and string field theory, Princeton University preprint (October 1985). S. Samuel, On the string field theory proposal of E. Witten, CERN preprint (February 1986). S.B. Giddings, The Veneziano amplitude from interacting string field theory, Princeton University preprint, January 1986. E. Cremer and J.L. Gervais, Nucl. Phys. B 76 (1974) 209. S. Mandelstam, Nucl. Phys. B 69 (1974) 77. D.J. Gross and A. Jevicki, Princeton University preprint (May 1986). O. Bernier, N. Castel and D. Duhamel, La th~orie des champs de cordes de Witten, Rapport d'option, Ecole Polytechnique, Palaiseau (June 1986).