Sewing string tree vertices with ghosts

Sewing String Tree Vertices with Ghosts

Leonidas Sandoval Junior

Department of Mathematics Centro de Ciências Tecnológicas UDESC,

Universidade do Estado de Santa Catarina, Brazil

Received on 13 November, 2001

It is shown how to sew string vertices with ghosts at tree level in order to produce new tree vertices using the Group Theoretic approach to String Theory. It is then verified the BRST invariance of the sewn vertex and shown that it has the correct ghost number.

I Introduction

In the early days of String Theory, one way to obtain amplitudes for the scattering of an arbitrary number of strings was by using the factorization property, what means that the scattering amplitude of N strings may be interpreted as the scattering amplitudes of a smaller number of strings sewn together. This made it possible to build the N string scattering amplitude by knowing the expression for three string scattering amplitudes. Even though they were very ingenious and successful, those calculations didn't take into account the ghost structure of the vertices, and that is what is done here.

In [1], it was shown how to sew tree vertices without ghosts using the Group Theoretic approach to String Theory [2] in order to obtain a new, composite vertex. Following the same procedure, we shall calculate the scattering amplitude of N strings taking account the ghost structure.

We shall start with a short review of how to sew tree vertices without ghosts. What we must do is sew two legs of two vertices, one leg from each vertex. What we have in the beginning are two vertices V₁ and V₂ with N₁ and N₂ legs, respectively (Fig. 1).

We now sew leg E from V₁ with the adjoint of leg F from V₂. What we have now is the substitution of the two sewn legs by a propagator (Fig. 2). When this propagator is written in parametric form, it is an integration of one of the variables (in order to cancel one spurious degree of freedom) and a conformal factor which contains terms of L_n's acting on leg E only.

So the resulting vertex V_c (called the composite vertex) has the generic form

where the hermitian conjugate of

V

₂ is for the sewn leg

F only and

where

x is a suitable variable. In what follows, we shall often write

instead of

P, calling attention to the integration when necessary.

When the two vertices are sewn together, we identify legs E and F. We also identify the Koba-Nielsen variable z_E with one of the Koba-Nielsen variables of vertex , and the Koba-Nielsen variable z_F we identify with one of the Koba-Nielsen variables of vertex V₁. In [1], this identification is made in the following way: z_E may be identified with z_F-1 or z_F+1, and z_F may be identified with z_E-1 or z_E+1. So there are four possible combinations: a) z_E = z_F-1, z_F = z_E-1; b) z_E = z_F+1, z_F = z_E+1; c) z_E = z_F-1, z_F = z_E+1; d) z_E = z_F+1, z_F = z_E-1.

II Oscillator case

The two original vertices satisfy some overlap identities and so shall do the composite vertex. One particular overlap identity is given by considering the operator Q^mi with conformal weight d = 0 defined by [2]

where

are bosonic oscillators with commutation relations

The overlap identity is given by

Because we are considering the adjoint of leg F in vertex V₂, we must see what the adjoint of these overlap identities are. First, by the definition of Q^mi, we have that

So, the adjoint of the overlap equations is given by

We now take the overlap identity considering the effects of the operator Q^mi(x_i) on the vertex V₁ on a generic leg i and on leg E¹ (Fig. 3):

We may then insert the unit operator 1=

^-1 and multiply by without altering the result:

Since the conformal operator acts only on leg E, it will have no effect on Q^mi(x_i). In order to compute the effect of on Q^mE(x_E), we must know that, for a conformal transformation V acting on a conformal operator R(z) of weight d,

Since

Q

^mE(x

_E) has conformal weight

d = 0, we have

and (

Fig. 4)

The second term in the expression above is facing now leg F of vertex , or best, its Hermitian conjugate. Considering that the Hermitian conjugate of Q^mi (given by (7)), we then have the following overlap identity between legs i and F:

We can then make a cycling transformation in order to obtain the correct factor for an arbitrary leg

j (

j ¹

F) of vertex

. The only term that will be affected is the term depending on leg

F:

Doing this, the overlap identity for the composite vertex

V

_c (

Fig. 5) can be written as

which is the overlap equation between two arbitrary legs i and j of the composite vertex V_c.

But the overlap identity for the composite vertex, since none of legs i or j involves the propagator, must be given by

and so in order for the equation we have obtained for the overlap of the composite vertex

V

_c to be true we must have

which implies that the propagator is given by

In order to give an explicit expression for the propagator, we will now choose x_i to be of the form

This choice is called the "simple cycling" [1] and it is the one that simplifies our calculations the most. In this choice, the propagator is given by

or in terms of the

L

_n

^E operators

²,

This form works for all choices for the composite vertex discussed before³. The true propagator is given by expression (22) integrated over a suitable variable. Choosing this variable to be s = z_F - z_E, we then have

Before going any further, we must discuss another aspect of the theory that depends on the particular way in which the legs are identified during the sewing procedure. Let us consider the more general case of an arbitrary cycling V_i. This kind of cycling may depend on other coordinates that are not z_i. As an example, let us suppose that we are identifying coordinate z_E of leg E with coordinate z_F-1 of vertex and coordinate z_F of leg F with coordinate z_F-1 of vertex V₁. The overlap identity between legs i and E - 1 on vertex V₁ is

where we are calling

the cycling transformation on leg

j (

j = 1, ... ,

N). These cycling transformations may depend on the other legs. As an example when this happens, we take another choice of the cycling transformations x

_i that is not as trivial as (20) but gives a simpler formula for the propagator. This choice is given by [1][3]

which is the transformation that takes

z

_i-1,

z

_i and

z

_i+1 to ¥, 0 and 1, respectively. Its inverse is given by

In the case of the cycling given by (25), the cycling for leg E - 1 will depend on leg E, which is not present in the composite vertex. On this vertex, the overlap between leg E - 1 and an arbitrary leg j reads

This overlap equation involves terms that depend on leg

E, which is non-existent in the composite vertex

V

_c. The correct overlaps should be given by

where the cycling transformations

and

do not depend on legs

E or

F. So, in order to restore the correct cycling transformation for the composite vertex, a conformal transformation must be made on the cycling transformations on vertex

V

₁. These are given by

where

In this definition, we consider implicit that the transformation

= 1 since the conformal transformations on leg

E will not be part of the composite vertex and so need not be modified.

Considering the general case, we have that the overlap equation for V₁ (Fig. 6) that will lead to the correct composite vertex is now obtained from the original overlap

where x

_0i =

z and x

_0E =

z. By inserting conformal transformation (30), we obtain (

Fig. 7)

Inserting now the propagator, we obtain (

Fig. 8)

and we expect the composite vertex to have a different form (given shortly) than in (1) in order to amount for the contributions of the conformal transformations. The second term of the overlap is now facing leg

F of vertex

so that we have the following overlap between legs

i and

F:

We are now facing the conformal transformation C_F⁴, defined by

which is necessary in order to change x

_0F ® x

_F. Introducing this transformation we obtain (

Fig. 9)

Making a cycling transformation from leg F to leg j, we then obtain (Fig. 10)

Once again, a conformal transformation must be introduced because of the cycling transformations

. This is defined by

so that we now have (

Fig. 11)

The composite vertex must be defined in terms of the new cycling transformations and so it must now include the conformal transformations that perform this change. So, it will now be defined by

Considering this, the overlap identity for the composite vertex

V

_c can be written as

Since the correct overlap identity for the composite vertex is given by

we must have

which implies once again that

For the cycling transformation (25), it is only necessary to do conformal transformations on legs E - 1, E, E + 1, F - 1, F and F + 1, depending on the particular way the variables associated with these legs are identified with the variables associated to legs E and F. In this particular example (which is case a seen before), the conformal transformations are given by

where

For the cycling (25), the propagators obtained for the four possible combinations discussed before are given by

where

⁵

The true propagators are obtained when we integrate the expressions above multiplied by a suitable constant. The results are:

It is now necessary to verify the effect of the gauge transformations C₁ and C₂ on the composite vertex as given by formula (40). We shall do it by verifying the effect of C₁ on vertex V₁. In order to do this we need the explicit expression for the bosonic oscillator vertex V₁, given by [1] [4]

where

and

V

_0j are cycling transformations involving leg

E and the oscillators

have commutation relations given by (4, 5). Matrices

D

_nm(g) are defined in the following way [3]:

and have the following multiplication property:

In order to calculate the effects of the conformal transformation C₁ on these oscillators we must make use of the following conformal operator [2]

which has conformal weight one, what means it transforms like

An oscillator (n ³ 1) can be expressed in terms of this conformal operator in the following way:

Acting on it with the conformal transformation

C

₁, we have

Making now a change of variables x

_0i ® x

_i, we have

Expanding (

V

_ix

_i)

ⁿ in terms of

D

_nm(g) matrices and

P

^mi(x

i) in terms of the oscillators, we obtain

Performing the integrations we then obtain

Using the same process for

, we obtain

Using these transformation properties, the multiplication rules of matrices D_nm(g) (equation (56)) and the property [3]

we can show that the effect of the conformal transformation

C

₁ on vertex

V

₁ is given by

i.e. the effect of C₁ on vertex V₁ is to change ® and V_0j ® V_j thus eliminating the dependence of the cycling transformations of every leg except leg E on the latter. The same can be done to obtain the effect of conformal transformation C₂ on vertex , with the same results. So, the effect of these transformations is to eliminate from the cycling transformations of the composite vertex all dependence on the sewn legs E and F.

III Introduction of ghosts

We now introduce ghosts in the vertex so that what we must sew now are two vertices with some ghost variables attached to them, i.e. we will be considering the physical vertices [4] which have the correct ghost number. In this case, in addition to satisfying the overlap identities with the conformal operator Q^mi, the two physical vertices and the composite vertex must also satisfy some overlap identities with the conformal operators bⁱ and cⁱ, given by [4]

where and are ghost anticommuting oscillators with anticommutation relations

These operators have, respectively, conformal weights 2 and -1, what means that they transform like

The overlap identities for a vertex V with these operators are given by

We shall be working here with overlap identities for the physical vertex U [4], which has the correct ghost number, instead of the overlap identities for vertex V. The physical vertex is given by [4]

where a, b, c are any three legs of the vertex and the matrix is given by

where the cycling transformations are now defined on the complete generators

of the conformal algebra of the bosonic oscillators and of the ghost oscillators. These vectors

have the following property:

In order to derive the overlap identity for the physical vertex U, we must multiply the overlap identity for V by the same factor as in equation (74),

and pass it through the overlap identities, obtaining

From (80) we can see that there will be an anomalous term in the cⁱ overlap of the physical vertex U unless both legs i and j are precisely those legs (a, b or c) that do not have any ghosts attached to them. These ghosts which are attached to all the other legs are responsible for the anomalous terms.

III.1 Analysis of the ghost number

Before going any further, it is necessary to make some considerations on the ghost number of the composite vertex. As we shall see shortly, in the case where we perform the sewing with ghosts included, using the physical vertices, the resulting composite physical vertex will not have the correct ghost number unless we insert some extra ghosts in vertex U₁ before the sewing takes place. Considering this, we shall define the composite vertex to be given by

where

G are some extra ghosts that will be introduced in order to make vertex

U

_c have the correct ghost number and

P is the propagator (in its integrated form).

We must now analyze the ghost number of the composite vertex and of its parts in order to calculate the ghost number that the extra ghosts G must have. In order to do this, we shall use the ghost number operator N_gh. For a vertex with N legs, the ghost number operator is defined by

The reason why the ghost number operator is a sum from

i = 1 to

i =

N is because there are

N vacua that will annihilate the operators corresponding to each one of them. When acting on the physical vertex, this operator gives a ghost number

N, what is the correct ghost number for a tree vertex with

N legs.

In the case of the composite vertex, it has N₁+N₂-2 legs (because it does not have legs E and F, which have been sewn together) and so it must have ghost number (N₁ + N₂ - 2). For this vertex, the ghost number operator can be divided into two parts:

where

This ghost number operator will have the following effect on the composite vertex:

Given formula (81) for the composite vertex, we then have

In order to calculate this, we must pay some attention to terms one and three of the right hand side of the expression above. We know that

where

Taking the Hermitian conjugate of equation (89) only on leg

F, we obtain

Since for

(and for any arbitrary ghost number operator)

we then obtain, substituting (88), (92) and (93) into equation (87),

Passing

through the extra ghosts

G, we then obtain

We must now remember that, in the composite vertex, we identify every operator on leg E with operators on leg F so that = . Doing this, the last two terms in (95) cancel and we obtain the following result:

The fact that U_c has ghost number N₁ + N₂ - 2 then implies that

i.e. the extra ghosts that must be introduced in vertex

U

₁ must have ghost number 1.

⁶

There is an infinite number of combinations of ghosts that have ghost number 1. We could have any linear combination of ghosts of the type b, bcb, bcbcb, etc. but it will prove to be simpler to choose G to be a combination of b¢s only so that we may represent it as

where

are the coefficients of the linear combination. In order to determine the correct linear combination, we must use some other conditions, like BRST invariance of the scattering amplitude. This we shall see next.

III.2 BRST invariance

We must now impose that the scattering amplitude obtained from the composite vertex is BRST invariant and check whether this condition is strong enough to determine G. The scattering amplitude [1] is obtained by acting with the composite vertex

on a certain number of physical states (|c

₁ñ|c

₂ñ ... |c

_Nñ) and then by integrating over all variables

z

_i (

i = 1, ... ,

N

₁ +

N

₂;

i ¹

E,

F):

P is the propagator in its integrated form and

G are the extra ghosts to be inserted in

U

₁.

The action of the BRST charge Q on this scattering amplitude is given by

The first and third terms will result in total derivatives that give zero when one integrates over some variables [1] and so what remain are just the second and third terms.

The commutator P is given by a pure conformal transformation, and it is a function of the generators (n = -1, ...) only. As the BRST charge commutes with all 's, i.e.

we have

Considering now that

we then have, for

G given by (98),

In order for the scattering amplitude W to be BRST invariant, expression (105) must be zero or a total derivative (that can be integrated out to become a null surface term). At the same time, we want these extra ghosts to place (talking in terms of the simple cycling) a ghost on one of the legs in U₁ that do not have any ghosts attached to them. If we now remember property (76), we see that we can satisfy these constraints in a nice way by choosing G to be given by

where

a (

a ¹

E) is one of the legs of vertex

U

₁ that does not have ghosts attached to it. Inserting these ghosts in vertex

U

₁, we have

Using formula (106) for the extra ghosts G, we then have

what is a total derivative that will vanish when one does the integration over

z

_a.

In references [5] and [6], the extra ghosts have been placed in the propagator. Although this can be done, there is no way one can derive a formula for the ghosts in the propagator for a general cycling. In that case, the extra ghosts must be derived and BRST invariance has to be checked for each particular cycling. Also, the resulting composite vertex obtained in that case is not similar in its ghost structure to an ordinary tree vertex, although it has the correct ghost number.

III.3 Overlap identities

We must now use the overlap identities to determine the propagator that satisfies them. In order to do this we shall start with vertex , which is the vertex with cycling transformations which involve leg E. Considering equations (79) and (80), the overlap identities for vertex between an arbitrary leg i and leg E are given by (Fig. 12)

The extra ghosts must then be inserted in vertex so that the composite vertex will have the correct ghost number. Multiplying expressions (109) and (110) by the extra ghosts G (given by (106)) and passing them through the overlaps we obtain (Fig. 13)

The second and third terms of equation (112) can be combined so that it becomes

At this point, we must introduce conformal transformations of the type of C₁, given by (30) in order to have at the end the correct cycling transformations for the composite vertex. In order to do this we need to use matrices E_nm(g), defined by [5]

which have the following properties:

The action of the operator C₁ on the ghosts can then be calculated in the following way: first we write in terms of an integral over the conformal operator (x_i)

Then we insert the operator

C

₁:

After a change of variables x

_i =

V

_0ix

_0i we have

⁷

Using matrices

E

_nm(g), we then may expand x

_0i in terms of x

_i. If we also expand

b

ⁱ(x

_i), we then obtain

Performing the integration we then have

Using (123) in equations (111) and (113) and multiplying (111) by ( dx_i/dx_0i) ^-2 and (113) by dx_i/dx_0i, we then have (Fig. 14)

Before going further, some words must be said about the effects of on vertex with the extra ghosts G. This is given explicitly by [4][5]

Making use of matrices F_nm(g), defined by [5]

we may calculate in a similar way as we did for the

ghosts the effect of

C

₁ on the

ghosts, obtaining

Using this together with the property

and equation (123), we may then show that the result of acting with

on

G is

So we can see that in this case the action of

on

G is not just to change

V

_0i ®

V

_i. Because of the peculiar nature of

, it transforms as

Only in one particular group of cycling transformations (as we shall see later) will this be just equivalent to changing

V

_0i ®

V

_i. We shall call from now on

G

º

U

₁ and

º

V

₁. The calculation for vertex

C

_F

will be similar to the one we have just made for

G

.

Having done this, we must insert the propagator into the overlap identities (124) and (125) in the same way as in the case with no ghosts. But now we must take extra care since there are terms depending on in the second term of equation (125). Using equation (123) as a guideline, we have

so that the result of inserting

into overlaps (124) and (125) is (

Fig. 15)

The operators of the overlap equations are now facing leg F of vertex . In order to obtain the overlap identities for this leg, we must now identify the operators of leg E with the ones of leg F, which are adjoint operators:

First, as

b

^F and

c

^F are conformal operators with weights 2 and -1, respectively, we have

Then, for

and

, we obtain

Then, we must also make the change

So, the overlap equations become

Formula 141-142

We are then facing the conformal transformation C_F that takes x_F into x_0F. Inserting this transformation we obtain (Fig. 16):

We are facing now vertex

. This vertex satisfies the following overlap identity [4]:

Using equation (117), we have

All terms on

with

t ³ 2 get annihilated by the conjugate vacuum |0ñ

_F, while we may use identity (145) to substitute the terms in

,

r = -1, 0, 1. Doing this, equation (144) becomes

Equations (143) and (147) are the overlap identities between legs

i and

F.

In order to obtain the overlaps between leg i of vertex U₁ and an arbitrary leg j of vertex , we must now perform a cycling transformation that will take the operators from leg F to leg j. The effect of this transformation on c^F(G^-1V_ix_i) is

Then, we must also write

Doing this, overlap equations (143, 147) become (

Fig. 17)

The operators are now facing the ghosts that surround vertex

(like in equation (74)):

so that we must insert these ghosts into the expressions for the overlaps. Before doing that, we must notice that the extra ghosts acting on vertex

have at their left both the conformal transformation

C

_F and the propagator

so that we must first pass them through in order to reach vertex

U

₁:

Then, identifying legs

E and

F, we have the following expression for the ghosts:

We may now pass it through the conformal transformation

C

₁, obtaining

This is now facing vertex

, which satisfies the following overlap identity [4]:

Using this identity, we may then write the extra ghosts as

Passing it back through

C

₁ we then have

We now insert these ghosts into expressions (150) and (151). We do so by multiplying them by (158) and passing it through the first term of the overlaps. What we obtain is (Fig. 18)

We are then facing the last term of this composite vertex: the conformal transformation C₂ on vertex . Inserting it into equations (159) and (160), we obtain (Fig. 19)

where we have called C_Fº and C_Fº .

We must now extract the ghosts from vertex U₁ in the second term of equation (162):

Identifying

with

and passing now these ghosts through the conformal transformation

C

_F, we obtain

Using now overlap identities (145) for vertex

and inserting the conformal transformation

C

₂, we then obtain

Substituting (165) into equation (162) we then obtain the overlap identities between legs

i and

j:

If we now impose that these are the correct overlap equations between legs i and j of the composite vertex we then must have:

what fixes the propagator as

which is the same form of the propagator for the bosonic part, but now with the cyclings defined on the complete generators

. The overlap equations now read

These are the overlap identities between legs

i and

j of the composite vertex

U

_c.

From these overlap equations it is then possible to derive the form of the composite vertex. It is given by

Formula 172

Although this is the correct composite vertex for a general cycling, its ghost structure is not very apparent. We may use the explicit expression for the vectors

[7]:

where

and

If we assume now that the cycling transformations

and

do not depend on the variables

z

_i of vertex

U

₁ or

z

_j of vertex

, and if we have in mind that

= 0 for any leg

i of vertex

U

₁ and a leg

j of vertex

(or vice-versa), then we have

what simplifies things considerably. We shall call all cyclings that have such properties "simple cycling-like". In this kind of cyclings, each leg has its own ghost attached to it, with the exception of three of the legs which have no ghosts attached to them. We will now consider three cases separately: one in which none of the legs

E or

F have ghosts attached to them, one in which one of these legs (say

E) has a ghost attached to it, and a case where both legs (

E and

F) have ghosts attached to them.

The composite vertex for the case where neither E nor F (we choose E = b and F = g) have ghosts attached is given by

where

is the vertex given by the first two terms of equation (172). For the case where leg

E has a ghost attached to it, but not leg

F (we shall call

F =

g), the composite vertex is given by

In the last case, where both

E and

F have ghosts attached to them, we then have

We can see that, for each case, the composite vertex has the correct ghosts number (

N

₁ +

N

₂ - 2).

IV Conclusions

Using overlap identities, two vertices were sewn together in order to become a composite vertex. The calculations have been done with the correct ghost numbers for each vertex and the result has both BRST invariance and the correct ghost counting.

Acknowledgements

This work has been done in the Department of Mathematics, King's College London, University of London, under the supervision of Professor P. C. West, and finished at the University of São Paulo (USP). The author wishes to thank Professor West for suggesting, guiding and correcting this work and Mrs. Nanci Romero for helping with the figures. My gratitude to CAPES and CNPq, Brazil, for financial support. G.E.D.!

References

[1] A. Neveau and P. West, Cycling, twistingand sewing in the Group Theoretic Approach to Strings, Comm. Math. Phys. 119, 585 (1988).

[2] A. Neveau and P. West, Group Theoretic Approach to the open bosonic string multi-loop S-matrix, Comm. Math. Phys. 114, 613 (1988).

[3] P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, N-point g-loop vertex for the free bosonic theory with vacuum charge Q, Nucl. Phys. B322, 317 (1989).

[4] M. Freeman and P. West, Ghost vertices for the bosonic string using the Group Theoretic approach to String Theory, Phys. Lett. B205, 30 (1988).

[5] P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, N-point g-loop vertex for a free fermionic theory with arbitrary spin, Nucl. Phys. B333, 635 (1990).

[6] P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, A simple expression for the multiloop amplitude in the bosonic string, Phys. Lett. B199, 49 (1987). P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, n-string vertex and loop calculation in the bosonic string, Nucl. Phys. B298, 527 (1988).

[7] L. Sandoval Jr., String Scattering, Ph.D. thesis (1995) King's College London, Univ. of London (excerpts available with the author).

²Other forms for this propagator are given by [1]:

³This affirmation is usually not valid for other choices of x_i.

⁴In this case, like we have seen for leg V_E, V_F may depend on the variable z_E.

⁶This contrasts with references [5] and [6] where it is claimed that the extra ghosts should have ghost number 1, 2 or 3, depending on the way one chooses legs E and F to have or not to have ghosts attached to them.

⁷Note that, because x_i is a polynomial in x_0i (with no constant term), then x_0i = 0 Þ x_i = 0.

¹

As we shall be seeing soon, this form of the overlap will not lead to the correct composite vertex in the case where the cycling transformations of the legs that are not sewn involve the sewn legs

E or

F.

⁵

The coefficient

a can be connected with the coefficient

c in reference [1] by

Publication Dates

• Publication in this collection
  26 Nov 2002
• Date of issue
  Mar 2002

History

• Received
  13 Nov 2001