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Operator construction of ghost g-vacuum, handle operator and BRST-invariant multiloop amplitude of the bosonic string
Volume 221, number 3,4
PHYSICS LETTERS B
4 May 1989
OPERATOR CONSTRUCTION OF GHOST g-VACUUM, HANDLE OPERATOR AND BRST-INVARIANT MULTILOOP AMPLITUDE OF THE BOSONIC STRING Ursula C A R O W - W A T A M U R A , Zyun F. EZAWA, Akinorl T E Z U K A Department of Phystcs, Tohoku Umverszty,Sendal 980, Japan and Satoshi W A T A M U R A Department of Physics, College of General Educatzon, Tohoku Umverstty, Sendat 980, Japan Received 26 January 1989
Applying the Feynman-hke rules m the operator formahsm w~thbosomzed bc-ghost, we construct the g-loop N-string vertex out of three-string vertices and propagators. The measure factor is analyzed in detail. It is shown that a part of the ghost contnbutmn can be represented by using g/2-dlfferentlal and Rlemann constants The resulting measure factor is represented by the ghost gvacuum w~ththe b-ghost insertion accompanied by a contour integration This contour integration can be performed exphcltly in the one-loop case, in which we reproduce the well-known formula of the one-loop N-tacbyon amplitude
In o r d e r to i m p r o v e our c o m p r e h e n s i o n o f the theory o f the string it is i m p o r t a n t to investigate the structure o f the m u l t i l o o p a m p l i t u d e from various points o f view. One recent aspect which has revealed some m a t h e m a t ical structure o f the string a m p l i t u d e is its deep connection with the algebraic geometry. On the other hand, from the physical p o i n t o f view, the string theory can be u n d e r s t o o d as the theory o f an infinite n u m b e r o f fields, i.e., the dual resonance model. In the dual resonance m o d e l we can construct the multiloop a m p h t u d e by a simple o p e r a t o r algebra. These approaches show us c o m p l i m e n t a r y aspects o f the string theory. The a p p r o a c h taken here is the use o f the Feynman-llke rules as originally d e v e l o p e d in the dual resonance model [ 1-3 ]. In this a p p r o a c h the basic objects, i.e., three-string vertex [2,4,5 ] and the propagator, are completed by adding the ghost contribution to ensure the BRST invariance ~ . We apply the F e y n m a n - l i k e rules presented in ref. [ 31 ] with Lovelace's p a r a m e t r i z a t i o n [ 1 ] to construct the multiloop a m p h t u d e . We analyze especially the ghost part and the measure factor in detail. In our formulation, the bc-ghosts are bosonized. The bosonized formulation is necessary when generalizing our m e t h o d to the superstring case, especially for including the R a m o n d sector. Actually, we have already o b t a i n e d the multlloop gv a c u u m o f the m a t t e r part o f the superstrlng in terms o f 0-functions by using the bosonlzed formalism [ 30 ]. F o r the ghost case, which we shall analyze in this letter, we expect new terms including the g/2-differential describing the effect o f the b a c k g r o u n d ghost charge [32]. On the other hand, in the bosonlzed formulation the b-ghost insertion which is necessary to cancel the background ghost charge, is m a d e into the p r o p a g a t o r together with a contour integration. As we shall see, this creates some technical subtlety when evaluating the multiloop amplitude. However, the analogy with the b-ghost inser~ A number of papers have been presented to reformulate three-stnng vertices in a BRST lnvarlant form [6-9 ], and further developments have been made to extend to the N-stnng vertices including the superstrmg case [9-18], The diagrammatic approaches based on the BRST invariance are considered in refs. [ 19-23 ] See also refs. [24,25 ] The construction of the BRST lnvanant multlloop amphtude in such an approach has been investigated for both the bosomc and the supersymmetnc stung by many authors, see for example refs [ 16,19-22,26-31 ] 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
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tion with Beltrami differentials appearing in the algebraic geometrical approach becomes more transparent [ 31 ]. Following the notations introduced in ref. [ 31 ], we write the three-string vertex as a bra-type state (V 31 and the propagator as a ket-type state ] P), where the propagator contains the measure factors as well as the b-ghost insertmn. The g-loop N-string vertex can be constructed by sewing these vertices by propagators according to the Feynman-like rules. For this end, it is convenient first to make a tree (2g+N)-string vertex, (vtr~'2g+NI, and then connect pairwlse its 2g legs by propagators. Such a vertex is represented as
(W"°°P'NI=
2g+N-- 2 3g+N-- 3
17]
I-I
t=l
t=l
g
(V'31p,) = l-I (Vtree'2g+NlP,),
(1)
t=l
where suffixes t and z distmguish the vertices and propagators. We choose the (2g+N)-string vertex of the peripheral type for (vtree'2g+NI, which IS given by [ 31 ]
£dy, (vlree'2g+NI = f dU2g+N2g+N--3 [I d 2hi F : r e e ( y , ) ( ~tree,2g+N; {y} 10)
(2)
l=l
where Fttree ( y , ) = (Yt--ZI)(Y'--Z~+2) Zl - - Z , + 2
d V 2g+N
~
]-I2g+NdZs(Za--Zb)(Zb--Zc)(Zc--Za) 2g+N I-[ r= 1 (Zr--Zr+l)
(3,4)
with the convention 2 g + N + r = r. Since we are using Lovelace's parametrlzatlon, the three vertex satisfies duality up to a gauge transformation co [ 19,31 ]~2. The operator part ( ~ree,2g+N; {Y) I in eq. (2) is the symmetric (2g+N)-string vertex with b-ghosts inserted, derived in ref. [ 11 ], and can be represented by using the string emission operator ~F~ [ 12 ] as (3utree,Xg+N; {y}]__ / f ' I I Y ' ( I ) ' y ' ( 2 ) . . .y~(2g) --AX"I--AA,~AIA2 . A2g-la
2g+N--3
17 t=l
b(B)(yz)~..(B~+l)~..(2g+2) -~,~(2g+S) B,B~ .... _ , ~ l0 )B~.
(5)
In terms of the bosonized ghost a, the ghost part of the string emission operator ~'g(~)o~t,ABIS defined as ~3 ~(r)
__ { gh°~t'AB--~(q=01B(q=01d(a~0+O~'~d--O~'0):exp
~nl a(m ( w ) l n ( W - - Y r ( Z ) ~ o I T ( r , ( z )
~ d2 ~ Owl
co
~nl
co
\ x/ OYr( z ) "/
+Ow[cr(A)(1/w)+Qln(1/w) ] ln(\x/0F(w)0Y,.(z Y'(z) )Oa(~)(z) +Ow[a(A)(1/w)+Qln(1/w)] ln(1-wz)Oa(B'(z)]}:
Iq=0)A.
(6)
The superstring (r) labels the external string while the indices A, B denote the internal strings. Q denotes the background ghost charge, and here Q = - 3. Each string emission operator is defined such that when we multiply an arbitrary on-shell state onto the external string (r) it reduces to the vertex operator for the emission of the corresponding state. The projective transformation Yr(Z) assocmted with its legs (r) is
Yr(z)=(;
1
oo ) ,
(7)
Zr+ 1 Zr-- 1 #2 The gauge factor o9 is the projective transformation with fixed points 0 and 1 and depends on the way of the construction of the tree a m p h t u d e This factor is harmless since at commutes with the BRST charge and becomes the identity when external legs are coupled with physical states, or when a leg is sewn w]th the twisted propagator to make a loop, 1 e , o9 Cs)[p h y s ) ~= [p h y s ) s and o9 (s) [ p ) . r = ] P)sr ,3 Our bosonizatlon formula is b = e - ° and c = e * [ 33 ] Since the main concern of this letter is the ghost contribution, here and hereafter we give expllot formulae only for the ghost part The details of the matter p a n can be found m ref. [ 30 ].
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in Lovelace's notation [ 1 ], and the transformatmn F is the inversion, F ( z ) = 1/z. Having constructed a tree ( 2 g + N ) - s t r m g vertex by string emission operators, a multiloop vertex is obtained by connecting the external legs of the tree vertex by propagators as indicated in fig. 1. The propagator I P) which connects the rth and sth string leg is defined with a b-ghost insertion and the Chan variable x as IP)=
f
dx x(1-x)
f dff ~ ~(1-~)b(r)({)l~>rs"
(8)
Note that the b-ghost can be inserted into that leg of the propagator which is attached to the sth leg instead of that attached to the rth leg, since the propagator is symmetric as shown in ref. [31 ]. The contour integration with respect to ( is to be performed around the propagator. The operator part I ~ > rs in eq. (8) is the direct product of ghost and matter contribution. The ghost part is given by [ ~ghost )rs = :exp[ co ~ 2--~i dz ~ ~~Y~iO~[a(r)(1/z)+Qln(l/z)] ln(F'(z)-y'~O co \ ~ J Y[a(~)(1/Y)+Qln(1/Y)]: X O(a(~,~ + a(~s~)[q=O)r [ q = O ) s ,
(9)
where
0' ? )
.
The advantage of introducing the string emission operator is that we can calculate the operator part of each loop separately as an independent building block for constructing multlloop amplitudes. We call this building block a handle operator ~2Ac, which is defined by
a,c-r~r~(
-
l r - I ~ >rs.
(l l)
where ( - 1 ) f = exp (rcio~.~d) which generates the proper spin structure for the ghost loop. After some calculation using the coherent state method [ 3 ], we obtain
(~)g-I°°o I
2g
IAN )
N
Fig. I. The tree (2g+N)-stnng vertex of the penpheral type is represented by using (2g+N) stnng emission operators. Then, the g-loop N-stnng vertex xs constructed from xt by connecting palrwlse 2g external legs by propagators. The g-loop part is descnbed by the g-loop operator (og-t°°PI, which is a product of g handle operators. The operator IAs) descnbes the contnbutmn of the external legs. 301
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PHYSICS LETTERSB
~,-2ghost,AC = C ( q = 0 i ~ ( O/A,0
-ao,0 C + Q ) ( d e t C l-loop)
4 May 1989
1
X:O[001( B ~2~11 {Oa~C)(z)(°(z'z°)+O:[a(A)(1/z)+Qln(1/z)]q~(F(z),zo)}+QA) Co
ln(E(x'Y)~
×exp{ ~ dx Co
Co
+½0x[(7(A)(1/x)+Qln(1/X)]Oy[a(A)(1/Y)+QIn(1/Y)] ln(xX--@_yE(F(x),F(Y))) + a a ( ° (x)0v [a(A) (1/y)
+ Q In(1/y)
] In yE(x,
F(y) )]
+Q -~-~ni{Oz[a(A)(1/z)+Qln(1/z)]lnz-lSl(1/z)+OatC)(z)lnSl(z)}
}
(12)
:lq=0)A.
Co
where det Cl-~ooo=
fi
B = ~ 1i In
(l_Kn),
n= 1
(o(Z, Z o ) = 2 ~ i l n J=½(B+I),
fi
k=--oo
fi ( T k+~(a),T(b),Tk(a),b),
(Tk(z),T(a),Tk(zo),a),
S,(z)= fi
(13,14)
k = --oo
E(x,y)=(x-y) fi (T~(x),y, Tk(y),x), n=l
(T"(z),f'(O),T"(O),T-IF(O)).
(15,16) (17,18)
n=0
Here, the transformation T(z) which gives the Mrbius transformation generating the Schottky group of the handle and K in eq. ( 13 ) is its multiplier. T(z) is defined by
T= yfl-,y;-i, where Y,(z) and Y,(z)
(19)
are the projective transformations assocmted with the two legs sewn by the propagator. The quanUties B, (0(z, Zo) and E(x, y) are the period matrix, the first abelian integral and the prime form for the one-loop case in the Schottky parametnzation, respectively. The quantity A is the Riemann constant and S~ (z) can be identified with the ½-differential with no zeros or poles in the fundamental region for the genusone surface [ 34,35 ]. The latter can be shown by investigating the transformation properties and pole structure. Hence, each term in the handle operator has a geometrical meaning. After sewing the 2g legs by g propagators (fig. 1 ) and shifting the inserted b-ghosts to the internal string [ 31 ], the g-loop N-string vertex ( 1 ) becomes (vg-lo°p.NI
= dVZg+NE2,(1--2,),=,J2-mm s
~
I,-_rIlb(YD
,0
b(y')IAN)' (20)
where IA N) denotes the contribution of the N external strings: "y, (2g+ 1 ) ~e, (2g+ 2)
[AN>B =--BBI
302
--BIB2
¥-(2g+ N)]O)B
. . . . BN-IBN
N
(21)
Volume 221, number 3,4
PHYSICS LETTERS B
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and the (g2g~°°p I is the g-loop operator given by the product o f g handle operators A~(-Qg-'°°PI --A (OI-QaA,-QA~A~...I2A~_,A,.
(22)
This operator can be identified with the g-vacuum of the bosonlzed ghost. The c-number function F~°°P(37~) in eq. (20) is originated from the factor ( ( 1 - ~ ) in the propagator (8) which is used to make the sth loop. Although the b-ghost insertion is symmetric with respect to the propagator leg into which it 1s made, the function F ] °°p (37~) is different. Depending into which leg the b-ghost is inserted, it is either
(J2s-Z2s-1)(Jls-Z2s) ,
F?°p(jTs) =
(23)
Z2s-- 1 -- Z2s or
F~opOTs) =
(Y;,-z2,)(Ys -ZEs+l )
(24)
Z2s -- Z2s + I However, after the contour integration it turns out that both choices The physical amplitude can be obtained by saturating IPhys) - E r Imatter )(r)®11; ghost)(r), where I 1; ghost )(r) denotes script (r) distinguishes the various legs. With this physical state, the (vgq°°p'Nlphys)
=
f
~'*rl][/lg'l°°p'N/Og'l°°PN ~-matter
I 2g+N F[ Vmatter 0 ) ( r )i
give the same result, as is expected. the external legs by physical states the ghost-number-one state and the superg-loop N-particle amplitude is
.
r=2g+ 1
(25)
where \/ .Ogl°°P . . . tter I is the matter g-vacuum given m ref. [30], and --matterlf(r) IS the ordinary vertex operator o f the matter part with conformal weight one. All contributions from the ghosts are included in the measure dM g~°°o'N given by
dMg-I°°p'N-~dr2g+Nt~___l = g
dxt :~,(1-.~,)
2g+N-3 ~ ~if: re e ( yt ) ,=,
2g+N
2g+N--3
×(g2g~°~Pl I ~ b ( L ) s= 1
I-I
b(y,)
t= I
fi s=l
I]
c(z,)lO;ghost),
(26)
r=2g+ 1
We analyze this measure d M g~°°p'N in the following and show that it gives the correct measure for the oneloop case by evaluating it explicitly. The g-loop operator is obtained by contracting g of the handle operators as eq. (22) indicates. After a tedious but straightforward calculation, the result can be expressed as '~aghost I = (det c g l ° ° ° ) - ~( -- Qg[ :O
Bu~
~--~Oa(z)tPg-l°°P(z,zo)+QAv Co
~ni~iOa(x)Oa(y)ln(Eg'~,y))+Q Co
Co
~ dz
]
0a(z) In Sg(z)3: ,
(27)
Co
where ~/.tV R gl°°p is the period matrix, ~0gl°°p (z, Zo) the first abehan integral, and/g-loop (X, y) the prime form in the Schottky parametrlzation ~4. These terms in the exponent are obtained by the same methods as applied to the case of the g-vacuum of the matter part of the superstring [ 30 ]. The new terms characteristic for the contri#" Exphc~t formulae of these quantities as well as the c-number factor det C g-l°°p c a n be found mref. [ 30 ], where we have treated the matter case using the same constructmn. Note that the determinant det C g'l°°p lS also generated systematmally during the recurs~ve joining of the g handles. 303
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butxon of the ghosts are collected into the two terms proportional to the background ghost charge Q in eq. (27). It is easy to show using eq. (27) that the ghost operator part in eq. (26) is g
2g+N--3
~ ¢')g-loop
oo~o~ Ilb(Y,s) s=l
1-I b(y,)
t=l
2g+N
I1
r=2g+ 1
E
c(z,)lo> Z o ) - Z ~0g-'°°P(29s,Zo) s
×
Z~'~l°°p(y,,zo)+QA, i
Ils<,E(~,,)Tt)I-I,.,E07,,y,)l-[,
(28)
This should be compared with the expression of the ghost correlation function given from the algebraic geometry approach [32 ]. We expect that the quantities A, and Sg(z) can be identified with the vector of Riemann constants and the g/2-differentials, respectively, given in the Schottky paramemzatlon. For the one-loop case, the explicit form of A, and Su(z) are given m eqs. ( 17 ) and ( 18 ), and the idennficatlon has already been mentioned. For the two-loop case, the expression becomes already rather complicated and lengthy. And therefore, it is not easy to prove explicitly that they are the vector of Riemann constants and the g/2-differentxal for the multiloop case. The detailed analysis of these quantRies will be reported elsewhere. In this last part of the paper, we shall show that our formulation gives the well-known modular lnvariant result for the one-loop case. For this case, we can perform the contour integration in eq. (26) rather straightforwardly. The contour integrations for the peripheral tree part are done just as in the case of the tree amplitude [ 31 ]. After these integrations, there remains a single contour integration appearing in the handle, where the contour is taken around the a-cycle, i.e., the contour around the isometric circle [ 36 ]. Then, the measure in eq. (26) for the oneloop N-string becomes dMI"I°°p'N=dpH°°P'N l-I dzs
f~°°P(y),
(29)
S=3 d~.l_loop,N =
]-IU+t2dzr
dx
2(1-2)
(~--?])(~--Z2g+N)(Z2g+N--~)
_
drld~dzzg+N (z3-zl)(Zl-Z2)(z2-zl)
dKU+l ~- ~ dzr.
(30)
r~3
Here, ~ and r/are the fixed points, and K the mulnplier, of the projective transformation T(z) in eq. ( 19)#5. We have taken z~, zb and z~ in eq. (4) as the two fixed points q, ~ and the Koba-Nielsen variable z2 + N, respectively. In order to perform the contour integration in eq. (29), we have to evaluate the ghost operator part explicitly so that the pole structure of the amplitude becomes apparent. It ~s convenient to use the following form of the ghost g-vacuum which is equivalent to eq. (27) for g = 1: ~ 1-loop ,,/ ~gho~,
I=(detCl-l°°P)-lexp(-½inQ-¼inQ2B)
B
~-~mOa(z)~o(Z,Zo) Co
-~-~x~-~mOff(x)Oa(y)ln(~)-Q~miOa(z)ln(z-~)(Z-rl) Co
Co
1.
(31)
Co
Then, we get easily
~5The defimtlon of the transformation T(z) m terms of the fixed points and mulnpher is given by [T(z) -~/]/[T(z) -~] = K(z-q)/ (z-G) ( [KI ~<1) 304
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(')l-loop '~'ghos, Ib(y)c(z3)IO> ,0,[1/21
( (Z3 -- ~) (-73 -- 71) ~ -- Q/2
= (det C Hoop) - , e x p ( - ½inQ- ~i~Q2B) ~,t 1/2j (BI ~0(z3, )7) ) E(J3, z3) ~ (J7-~)07-~/)
,1
(32)
Since the prime form is expressed as O[ 1~1 (BI ~o(z3, 9) )
E ( z3, ?~) = x/ Otp(y )Otp( z3 ) 21rKl /8 ( de t C l-loop)3'
(33)
we obtain -r/)(z3-~)(~-r/) < g 2 ~ P l b ( p ) c ( z 3 ) l O > = ( d e t C l loop) 2 K1 ( z 3 ()~_/~) 2 (j~_ ~) 2
(34)
Consequently, performing the contour integration in eq. (29) we find that dKN+2
dMll°°p'N= -~
H
dzr(det C l-loop) 2 ( Z2+N -- q) ( Z 2 + N - ~)
(35)
By making use of the projective invariance we may choose that q=0,
~=oo,
Z2+N=I.
(36)
Then, eq. (35 ) reproduces the well-known formula for the measure of the one-loop N-tachyon vertex ~6. In this letter, applying the Feynman-like rules with Lovelace's parametrization [ 1 ], we have constructed the BRST invariant g-loop N-string vertex. The BRST invariance follows straightforwardly from that of the threestring vertex and the twisted propagator [ 31 ]. We have analyzed the m e a s u r e d m g-l°°p,N in detail which includes the ghost contribution. The ghost operator part in the bosonized form leads us to the ghost g-vacuum. First, we constructed a handle operator represented entirely in terms of the geometrical quantities, i.e. abehan integrals, period matrix, prime form, Riemann constant and ½-differential as in eq. (12). Then, the g-vacuum was generated simply by joining g handle operators. With this ghost g-vacuum, any ghost correlation function can be easily written down, see eq. (28), as in the case of the matter part [ 30 ]. The new terms characteristic to the ghost case are collected into the quantxties A, and Sg, which must be identified with the vector of Riemann constants and the g/2-differential, respectively. Our construction of the ghost g-vacuum seems to gives us the recursive construction of the g/2-differential out of ( g - 1 )/2-differential in the Schottky parametrization. Further studies on this problem are now in progress. One advantage of the operator formalism is that the factorization property is manifest by the way of construction which is not easy to see in the algebraic geometry approach. On the other hand, the modular lnvarlance of the multiloop amplitude is not evident from the construction and has to be shown separately. However, by completing the above discussed identifications, we can apply the results on the modular invariance obtained from the geometrical approach. As for the one-loop case, we have shown the complete agreement with the wellknown result and thus the modular invariance ts manifest. We would like to acknowledge Dr. K. Harada for helpful discussions. #6When taking the external legsas tachyons, the matter part in eq (25) becomes N+2 I-loop I l-[ exp[p,X(zs)]. I0) <~'~...... s=3
- ~
I]=, Ik~--~ ] kl-n-K) expl~,~,P'P" lnt(1/z'f~z%)exp{ [ln(zffz~)]2/2 In K}E(z., z.)] . 305
Volume 221, number 3,4
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PHYSICS LETTERS B
4 May 1989
OPERATOR CONSTRUCTION OF GHOST g-VACUUM, HANDLE OPERATOR AND BRST-INVARIANT MULTILOOP AMPLITUDE OF THE BOSONIC STRING Ursula C A R O W - W A T A M U R A , Zyun F. EZAWA, Akinorl T E Z U K A Department of Phystcs, Tohoku Umverszty,Sendal 980, Japan and Satoshi W A T A M U R A Department of Physics, College of General Educatzon, Tohoku Umverstty, Sendat 980, Japan Received 26 January 1989
Applying the Feynman-hke rules m the operator formahsm w~thbosomzed bc-ghost, we construct the g-loop N-string vertex out of three-string vertices and propagators. The measure factor is analyzed in detail. It is shown that a part of the ghost contnbutmn can be represented by using g/2-dlfferentlal and Rlemann constants The resulting measure factor is represented by the ghost gvacuum w~ththe b-ghost insertion accompanied by a contour integration This contour integration can be performed exphcltly in the one-loop case, in which we reproduce the well-known formula of the one-loop N-tacbyon amplitude
In o r d e r to i m p r o v e our c o m p r e h e n s i o n o f the theory o f the string it is i m p o r t a n t to investigate the structure o f the m u l t i l o o p a m p l i t u d e from various points o f view. One recent aspect which has revealed some m a t h e m a t ical structure o f the string a m p l i t u d e is its deep connection with the algebraic geometry. On the other hand, from the physical p o i n t o f view, the string theory can be u n d e r s t o o d as the theory o f an infinite n u m b e r o f fields, i.e., the dual resonance model. In the dual resonance m o d e l we can construct the multiloop a m p h t u d e by a simple o p e r a t o r algebra. These approaches show us c o m p l i m e n t a r y aspects o f the string theory. The a p p r o a c h taken here is the use o f the Feynman-llke rules as originally d e v e l o p e d in the dual resonance model [ 1-3 ]. In this a p p r o a c h the basic objects, i.e., three-string vertex [2,4,5 ] and the propagator, are completed by adding the ghost contribution to ensure the BRST invariance ~ . We apply the F e y n m a n - l i k e rules presented in ref. [ 31 ] with Lovelace's p a r a m e t r i z a t i o n [ 1 ] to construct the multiloop a m p h t u d e . We analyze especially the ghost part and the measure factor in detail. In our formulation, the bc-ghosts are bosonized. The bosonized formulation is necessary when generalizing our m e t h o d to the superstring case, especially for including the R a m o n d sector. Actually, we have already o b t a i n e d the multlloop gv a c u u m o f the m a t t e r part o f the superstrlng in terms o f 0-functions by using the bosonlzed formalism [ 30 ]. F o r the ghost case, which we shall analyze in this letter, we expect new terms including the g/2-differential describing the effect o f the b a c k g r o u n d ghost charge [32]. On the other hand, in the bosonlzed formulation the b-ghost insertion which is necessary to cancel the background ghost charge, is m a d e into the p r o p a g a t o r together with a contour integration. As we shall see, this creates some technical subtlety when evaluating the multiloop amplitude. However, the analogy with the b-ghost inser~ A number of papers have been presented to reformulate three-stnng vertices in a BRST lnvarlant form [6-9 ], and further developments have been made to extend to the N-stnng vertices including the superstrmg case [9-18], The diagrammatic approaches based on the BRST invariance are considered in refs. [ 19-23 ] See also refs. [24,25 ] The construction of the BRST lnvanant multlloop amphtude in such an approach has been investigated for both the bosomc and the supersymmetnc stung by many authors, see for example refs [ 16,19-22,26-31 ] 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
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tion with Beltrami differentials appearing in the algebraic geometrical approach becomes more transparent [ 31 ]. Following the notations introduced in ref. [ 31 ], we write the three-string vertex as a bra-type state (V 31 and the propagator as a ket-type state ] P), where the propagator contains the measure factors as well as the b-ghost insertmn. The g-loop N-string vertex can be constructed by sewing these vertices by propagators according to the Feynman-like rules. For this end, it is convenient first to make a tree (2g+N)-string vertex, (vtr~'2g+NI, and then connect pairwlse its 2g legs by propagators. Such a vertex is represented as
(W"°°P'NI=
2g+N-- 2 3g+N-- 3
17]
I-I
t=l
t=l
g
(V'31p,) = l-I (Vtree'2g+NlP,),
(1)
t=l
where suffixes t and z distmguish the vertices and propagators. We choose the (2g+N)-string vertex of the peripheral type for (vtree'2g+NI, which IS given by [ 31 ]
£dy, (vlree'2g+NI = f dU2g+N2g+N--3 [I d 2hi F : r e e ( y , ) ( ~tree,2g+N; {y} 10)
(2)
l=l
where Fttree ( y , ) = (Yt--ZI)(Y'--Z~+2) Zl - - Z , + 2
d V 2g+N
~
]-I2g+NdZs(Za--Zb)(Zb--Zc)(Zc--Za) 2g+N I-[ r= 1 (Zr--Zr+l)
(3,4)
with the convention 2 g + N + r = r. Since we are using Lovelace's parametrlzatlon, the three vertex satisfies duality up to a gauge transformation co [ 19,31 ]~2. The operator part ( ~ree,2g+N; {Y) I in eq. (2) is the symmetric (2g+N)-string vertex with b-ghosts inserted, derived in ref. [ 11 ], and can be represented by using the string emission operator ~F~ [ 12 ] as (3utree,Xg+N; {y}]__ / f ' I I Y ' ( I ) ' y ' ( 2 ) . . .y~(2g) --AX"I--AA,~AIA2 . A2g-la
2g+N--3
17 t=l
b(B)(yz)~..(B~+l)~..(2g+2) -~,~(2g+S) B,B~ .... _ , ~ l0 )B~.
(5)
In terms of the bosonized ghost a, the ghost part of the string emission operator ~'g(~)o~t,ABIS defined as ~3 ~(r)
__ { gh°~t'AB--~(q=01B(q=01d(a~0+O~'~d--O~'0):exp
~nl a(m ( w ) l n ( W - - Y r ( Z ) ~ o I T ( r , ( z )
~ d2 ~ Owl
co
~nl
co
\ x/ OYr( z ) "/
+Ow[cr(A)(1/w)+Qln(1/w) ] ln(\x/0F(w)0Y,.(z Y'(z) )Oa(~)(z) +Ow[a(A)(1/w)+Qln(1/w)] ln(1-wz)Oa(B'(z)]}:
Iq=0)A.
(6)
The superstring (r) labels the external string while the indices A, B denote the internal strings. Q denotes the background ghost charge, and here Q = - 3. Each string emission operator is defined such that when we multiply an arbitrary on-shell state onto the external string (r) it reduces to the vertex operator for the emission of the corresponding state. The projective transformation Yr(Z) assocmted with its legs (r) is
Yr(z)=(;
1
oo ) ,
(7)
Zr+ 1 Zr-- 1 #2 The gauge factor o9 is the projective transformation with fixed points 0 and 1 and depends on the way of the construction of the tree a m p h t u d e This factor is harmless since at commutes with the BRST charge and becomes the identity when external legs are coupled with physical states, or when a leg is sewn w]th the twisted propagator to make a loop, 1 e , o9 Cs)[p h y s ) ~= [p h y s ) s and o9 (s) [ p ) . r = ] P)sr ,3 Our bosonizatlon formula is b = e - ° and c = e * [ 33 ] Since the main concern of this letter is the ghost contribution, here and hereafter we give expllot formulae only for the ghost part The details of the matter p a n can be found m ref. [ 30 ].
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in Lovelace's notation [ 1 ], and the transformatmn F is the inversion, F ( z ) = 1/z. Having constructed a tree ( 2 g + N ) - s t r m g vertex by string emission operators, a multiloop vertex is obtained by connecting the external legs of the tree vertex by propagators as indicated in fig. 1. The propagator I P) which connects the rth and sth string leg is defined with a b-ghost insertion and the Chan variable x as IP)=
f
dx x(1-x)
f dff ~ ~(1-~)b(r)({)l~>rs"
(8)
Note that the b-ghost can be inserted into that leg of the propagator which is attached to the sth leg instead of that attached to the rth leg, since the propagator is symmetric as shown in ref. [31 ]. The contour integration with respect to ( is to be performed around the propagator. The operator part I ~ > rs in eq. (8) is the direct product of ghost and matter contribution. The ghost part is given by [ ~ghost )rs = :exp[ co ~ 2--~i dz ~ ~~Y~iO~[a(r)(1/z)+Qln(l/z)] ln(F'(z)-y'~O co \ ~ J Y[a(~)(1/Y)+Qln(1/Y)]: X O(a(~,~ + a(~s~)[q=O)r [ q = O ) s ,
(9)
where
0' ? )
.
The advantage of introducing the string emission operator is that we can calculate the operator part of each loop separately as an independent building block for constructing multlloop amplitudes. We call this building block a handle operator ~2Ac, which is defined by
a,c-r~r~(
-
l r - I ~ >rs.
(l l)
where ( - 1 ) f = exp (rcio~.~d) which generates the proper spin structure for the ghost loop. After some calculation using the coherent state method [ 3 ], we obtain
(~)g-I°°o I
2g
IAN )
N
Fig. I. The tree (2g+N)-stnng vertex of the penpheral type is represented by using (2g+N) stnng emission operators. Then, the g-loop N-stnng vertex xs constructed from xt by connecting palrwlse 2g external legs by propagators. The g-loop part is descnbed by the g-loop operator (og-t°°PI, which is a product of g handle operators. The operator IAs) descnbes the contnbutmn of the external legs. 301
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PHYSICS LETTERSB
~,-2ghost,AC = C ( q = 0 i ~ ( O/A,0
-ao,0 C + Q ) ( d e t C l-loop)
4 May 1989
1
X:O[001( B ~2~11 {Oa~C)(z)(°(z'z°)+O:[a(A)(1/z)+Qln(1/z)]q~(F(z),zo)}+QA) Co
ln(E(x'Y)~
×exp{ ~ dx Co
Co
+½0x[(7(A)(1/x)+Qln(1/X)]Oy[a(A)(1/Y)+QIn(1/Y)] ln(xX--@_yE(F(x),F(Y))) + a a ( ° (x)0v [a(A) (1/y)
+ Q In(1/y)
] In yE(x,
F(y) )]
+Q -~-~ni{Oz[a(A)(1/z)+Qln(1/z)]lnz-lSl(1/z)+OatC)(z)lnSl(z)}
}
(12)
:lq=0)A.
Co
where det Cl-~ooo=
fi
B = ~ 1i In
(l_Kn),
n= 1
(o(Z, Z o ) = 2 ~ i l n J=½(B+I),
fi
k=--oo
fi ( T k+~(a),T(b),Tk(a),b),
(Tk(z),T(a),Tk(zo),a),
S,(z)= fi
(13,14)
k = --oo
E(x,y)=(x-y) fi (T~(x),y, Tk(y),x), n=l
(T"(z),f'(O),T"(O),T-IF(O)).
(15,16) (17,18)
n=0
Here, the transformation T(z) which gives the Mrbius transformation generating the Schottky group of the handle and K in eq. ( 13 ) is its multiplier. T(z) is defined by
T= yfl-,y;-i, where Y,(z) and Y,(z)
(19)
are the projective transformations assocmted with the two legs sewn by the propagator. The quanUties B, (0(z, Zo) and E(x, y) are the period matrix, the first abelian integral and the prime form for the one-loop case in the Schottky parametnzation, respectively. The quantity A is the Riemann constant and S~ (z) can be identified with the ½-differential with no zeros or poles in the fundamental region for the genusone surface [ 34,35 ]. The latter can be shown by investigating the transformation properties and pole structure. Hence, each term in the handle operator has a geometrical meaning. After sewing the 2g legs by g propagators (fig. 1 ) and shifting the inserted b-ghosts to the internal string [ 31 ], the g-loop N-string vertex ( 1 ) becomes (vg-lo°p.NI
= dVZg+NE2,(1--2,),=,J2-mm s
~
I,-_rIlb(YD
,0
b(y')IAN)' (20)
where IA N) denotes the contribution of the N external strings: "y, (2g+ 1 ) ~e, (2g+ 2)
[AN>B =--BBI
302
--BIB2
¥-(2g+ N)]O)B
. . . . BN-IBN
N
(21)
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PHYSICS LETTERS B
4 May 1989
and the (g2g~°°p I is the g-loop operator given by the product o f g handle operators A~(-Qg-'°°PI --A (OI-QaA,-QA~A~...I2A~_,A,.
(22)
This operator can be identified with the g-vacuum of the bosonlzed ghost. The c-number function F~°°P(37~) in eq. (20) is originated from the factor ( ( 1 - ~ ) in the propagator (8) which is used to make the sth loop. Although the b-ghost insertion is symmetric with respect to the propagator leg into which it 1s made, the function F ] °°p (37~) is different. Depending into which leg the b-ghost is inserted, it is either
(J2s-Z2s-1)(Jls-Z2s) ,
F?°p(jTs) =
(23)
Z2s-- 1 -- Z2s or
F~opOTs) =
(Y;,-z2,)(Ys -ZEs+l )
(24)
Z2s -- Z2s + I However, after the contour integration it turns out that both choices The physical amplitude can be obtained by saturating IPhys) - E r Imatter )(r)®11; ghost)(r), where I 1; ghost )(r) denotes script (r) distinguishes the various legs. With this physical state, the (vgq°°p'Nlphys)
=
f
~'*rl][/lg'l°°p'N/Og'l°°PN ~-matter
I 2g+N F[ Vmatter 0 ) ( r )i
give the same result, as is expected. the external legs by physical states the ghost-number-one state and the superg-loop N-particle amplitude is
.
r=2g+ 1
(25)
where \/ .Ogl°°P . . . tter I is the matter g-vacuum given m ref. [30], and --matterlf(r) IS the ordinary vertex operator o f the matter part with conformal weight one. All contributions from the ghosts are included in the measure dM g~°°o'N given by
dMg-I°°p'N-~dr2g+Nt~___l = g
dxt :~,(1-.~,)
2g+N-3 ~ ~if: re e ( yt ) ,=,
2g+N
2g+N--3
×(g2g~°~Pl I ~ b ( L ) s= 1
I-I
b(y,)
t= I
fi s=l
I]
c(z,)lO;ghost),
(26)
r=2g+ 1
We analyze this measure d M g~°°p'N in the following and show that it gives the correct measure for the oneloop case by evaluating it explicitly. The g-loop operator is obtained by contracting g of the handle operators as eq. (22) indicates. After a tedious but straightforward calculation, the result can be expressed as '~aghost I = (det c g l ° ° ° ) - ~( -- Qg[ :O
Bu~
~--~Oa(z)tPg-l°°P(z,zo)+QAv Co
~ni~iOa(x)Oa(y)ln(Eg'~,y))+Q Co
Co
~ dz
]
0a(z) In Sg(z)3: ,
(27)
Co
where ~/.tV R gl°°p is the period matrix, ~0gl°°p (z, Zo) the first abehan integral, and/g-loop (X, y) the prime form in the Schottky parametrlzation ~4. These terms in the exponent are obtained by the same methods as applied to the case of the g-vacuum of the matter part of the superstring [ 30 ]. The new terms characteristic for the contri#" Exphc~t formulae of these quantities as well as the c-number factor det C g-l°°p c a n be found mref. [ 30 ], where we have treated the matter case using the same constructmn. Note that the determinant det C g'l°°p lS also generated systematmally during the recurs~ve joining of the g handles. 303
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butxon of the ghosts are collected into the two terms proportional to the background ghost charge Q in eq. (27). It is easy to show using eq. (27) that the ghost operator part in eq. (26) is g
2g+N--3
~ ¢')g-loop
oo~o~ Ilb(Y,s) s=l
1-I b(y,)
t=l
2g+N
I1
r=2g+ 1
E
c(z,)lo> Z o ) - Z ~0g-'°°P(29s,Zo) s
×
Z~'~l°°p(y,,zo)+QA, i
Ils<,E(~,,)Tt)I-I,.,E07,,y,)l-[,
(28)
This should be compared with the expression of the ghost correlation function given from the algebraic geometry approach [32 ]. We expect that the quantities A, and Sg(z) can be identified with the vector of Riemann constants and the g/2-differentials, respectively, given in the Schottky paramemzatlon. For the one-loop case, the explicit form of A, and Su(z) are given m eqs. ( 17 ) and ( 18 ), and the idennficatlon has already been mentioned. For the two-loop case, the expression becomes already rather complicated and lengthy. And therefore, it is not easy to prove explicitly that they are the vector of Riemann constants and the g/2-differentxal for the multiloop case. The detailed analysis of these quantRies will be reported elsewhere. In this last part of the paper, we shall show that our formulation gives the well-known modular lnvariant result for the one-loop case. For this case, we can perform the contour integration in eq. (26) rather straightforwardly. The contour integrations for the peripheral tree part are done just as in the case of the tree amplitude [ 31 ]. After these integrations, there remains a single contour integration appearing in the handle, where the contour is taken around the a-cycle, i.e., the contour around the isometric circle [ 36 ]. Then, the measure in eq. (26) for the oneloop N-string becomes dMI"I°°p'N=dpH°°P'N l-I dzs
f~°°P(y)
(29)
S=3 d~.l_loop,N =
]-IU+t2dzr
dx
2(1-2)
(~--?])(~--Z2g+N)(Z2g+N--~)
_
drld~dzzg+N (z3-zl)(Zl-Z2)(z2-zl)
dKU+l ~- ~ dzr.
(30)
r~3
Here, ~ and r/are the fixed points, and K the mulnplier, of the projective transformation T(z) in eq. ( 19)#5. We have taken z~, zb and z~ in eq. (4) as the two fixed points q, ~ and the Koba-Nielsen variable z2 + N, respectively. In order to perform the contour integration in eq. (29), we have to evaluate the ghost operator part explicitly so that the pole structure of the amplitude becomes apparent. It ~s convenient to use the following form of the ghost g-vacuum which is equivalent to eq. (27) for g = 1: ~ 1-loop ,,/ ~gho~,
I=(detCl-l°°P)-lexp(-½inQ-¼inQ2B)
B
~-~mOa(z)~o(Z,Zo) Co
-~-~x~-~mOff(x)Oa(y)ln(~)-Q~miOa(z)ln(z-~)(Z-rl) Co
Co
1.
(31)
Co
Then, we get easily
~5The defimtlon of the transformation T(z) m terms of the fixed points and mulnpher is given by [T(z) -~/]/[T(z) -~] = K(z-q)/ (z-G) ( [KI ~<1) 304
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(')l-loop '~'ghos, Ib(y)c(z3)IO> ,0,[1/21
( (Z3 -- ~) (-73 -- 71) ~ -- Q/2
= (det C Hoop) - , e x p ( - ½inQ- ~i~Q2B) ~,t 1/2j (BI ~0(z3, )7) ) E(J3, z3) ~ (J7-~)07-~/)
,1
(32)
Since the prime form is expressed as O[ 1~1 (BI ~o(z3, 9) )
E ( z3, ?~) = x/ Otp(y )Otp( z3 ) 21rKl /8 ( de t C l-loop)3'
(33)
we obtain -r/)(z3-~)(~-r/) < g 2 ~ P l b ( p ) c ( z 3 ) l O > = ( d e t C l loop) 2 K1 ( z 3 ()~_/~) 2 (j~_ ~) 2
(34)
Consequently, performing the contour integration in eq. (29) we find that dKN+2
dMll°°p'N= -~
H
dzr(det C l-loop) 2 ( Z2+N -- q) ( Z 2 + N - ~)
(35)
By making use of the projective invariance we may choose that q=0,
~=oo,
Z2+N=I.
(36)
Then, eq. (35 ) reproduces the well-known formula for the measure of the one-loop N-tachyon vertex ~6. In this letter, applying the Feynman-like rules with Lovelace's parametrization [ 1 ], we have constructed the BRST invariant g-loop N-string vertex. The BRST invariance follows straightforwardly from that of the threestring vertex and the twisted propagator [ 31 ]. We have analyzed the m e a s u r e d m g-l°°p,N in detail which includes the ghost contribution. The ghost operator part in the bosonized form leads us to the ghost g-vacuum. First, we constructed a handle operator represented entirely in terms of the geometrical quantities, i.e. abehan integrals, period matrix, prime form, Riemann constant and ½-differential as in eq. (12). Then, the g-vacuum was generated simply by joining g handle operators. With this ghost g-vacuum, any ghost correlation function can be easily written down, see eq. (28), as in the case of the matter part [ 30 ]. The new terms characteristic to the ghost case are collected into the quantxties A, and Sg, which must be identified with the vector of Riemann constants and the g/2-differential, respectively. Our construction of the ghost g-vacuum seems to gives us the recursive construction of the g/2-differential out of ( g - 1 )/2-differential in the Schottky parametrization. Further studies on this problem are now in progress. One advantage of the operator formalism is that the factorization property is manifest by the way of construction which is not easy to see in the algebraic geometry approach. On the other hand, the modular lnvarlance of the multiloop amplitude is not evident from the construction and has to be shown separately. However, by completing the above discussed identifications, we can apply the results on the modular invariance obtained from the geometrical approach. As for the one-loop case, we have shown the complete agreement with the wellknown result and thus the modular invariance ts manifest. We would like to acknowledge Dr. K. Harada for helpful discussions. #6When taking the external legsas tachyons, the matter part in eq (25) becomes N+2 I-loop I l-[ exp[p,X(zs)]. I0) <~'~...... s=3
- ~
I]=, Ik~--~ ] kl-n-K) expl~,~,P'P" lnt(1/z'f~z%)exp{ [ln(zffz~)]2/2 In K}E(z., z.)] . 305
Volume 221, number 3,4
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4 May 1989
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