# One-Loop Tachyon Amplitude

in Unoriented Open-Closed String Field Theory

###### Abstract

We calculate one-loop 2-point tachyon amplitudes in unoriented open-closed string field theory, and determine all the coupling constants of the interaction vertices in the theory. It is shown that the planar, nonplanar and nonorientable amplitudes are all reproduced correctly, and nontrivial consistencies between the determined coupling constants are observed. The necessity for the gauge group to be is also reconfirmed.

[3cm]KUNS-1575

hep-th/9905043

May, 1999
\recdateFebruary 14, 2021

## 1 Introduction

In previous two papers,[1, 2] which we refer to as I and II, we constructed a string field theory (SFT) for an unoriented open-closed string mixed system and proved the BRS/gauge invariance of the action at the ‘tree level’: namely, we have classified the terms in the BRS transform of the action into two categories, ‘tree terms’ and ‘loop terms’ and have shown that all the ‘tree terms’ indeed cancel themselves. And for the other ‘loop terms’, we have identified which anomalous one-loop diagrams they are expected to cancel.

It was left for a future work to show that those loop diagrams are indeed anomalous and the ‘loop terms’ really cancel them. However, the task to show this BRS invariance for general one-loop diagrams is technically rather hard and we have not yet completed it. Here in this paper, we address ourselves to an easier problem to compute the one-loop 2-point (open-string) tachyon amplitudes in our SFT. These one-loop amplitudes already suffer from a BRS anomaly and so the present calculation gives partially an affirmative answer to the above expected cancellation between the ‘loop terms’ and the one-loop anomaly. Moreover with this computation we shall confirm that the one-loop tachyon amplitudes are correctly reproduced in our SFT and can determine all the remaining coupling constants left undetermined in the previous work; in particular, we show that the gauge group must be .

The action of the present system, containing seven interaction terms, is given by

(1) | |||||

where , , , , and are coupling constants (relative to the open 3-string coupling constant ). For notation and conventions, we follow our previous papers I and II. The open and closed string fields are denoted by and , respectively, both of which are Grassmann odd. The multiple products of string fields are denoted for brevity as . The BRS charges with tilde here, introduced in I, are given by the usual BRS charges plus counterterms for the ‘zero intercept’ proportional to the squared string length parameter :

(2) |

The ghost zero-modes for the closed string are defined by , and .

The string fields are always accompanied by the unoriented projection operator , which is given by using the twist operator in the form , where for the open string case means also taking transposition of the matrix index. The closed string is further accompanied by the projection operator , projecting out the modes, and the corresponding anti-ghost zero-mode factor .

Among the seven vertices, the open 3-string vertex , open-closed transition vertex and open-string self-intersection vertex are relevant in this paper and have the following form:[2]

(3) |

The vertices here denoted by lower case letters are those constructed by the procedure of LeClair, Peskin and Preitschopf (LPP).[3] The are anti-ghost factors[4, 5, 6] corresponding to the moduli representing the two interaction points in the intersection vertex . Those LPP vertices generally have the structure

(4) |

where with momentum eigenvalue and the exponent is a quadratic form of string oscillators with Neumann coefficients determined by the way of gluing the participating strings. A point to be noted here is that since the gluing way depends on the set of the string lengths and, in our HIKKO type theory,[7] the string length is identified with the + component of string momentum ; i.e., for open string and for closed string, the exponent function has a nontrivial dependence on s. Note that and . Therefore the integration over in Eq. (4) is quite nontrivial. is the Fock bra vacuum of string

In the previous papers, we have shown that the theory is BRS (and hence gauge) invariant at ‘tree level’ if the coupling constants satisfy the relations

(5) | |||

(6) | |||

(7) | |||

(8) |

where the sign of has been changed from the previous papers I and II since we change the sign convention for the vertex in this paper by the reason as will be made clear in §5. These relations (5) – (8) leave only two parameters free, e.g., and , or and . We shall determine all the three parameters , and , thus giving a nontrivial consistency check of the theory.

The rest of this paper is organized as follows. First in §2, we show which diagrams contribute to the on-shell tachyon 2-point amplitude at one-loop level by using Feynman rule in the present SFT. To evaluate the amplitudes of those diagrams explicitly we need to conformally map those diagrams into the torus plane and to compute the conformal field theory (CFT) correlation functions on the torus. We present in §3 such conformal mapping for each diagram explicitly, and calculate in §4 the Jacobians for the changes of moduli parameters associated with the mappings. In §5 we first give some discussions on the ‘generalized gluing and resmoothing theorem’ (GGRT)[3, 8] to fix the normalizations of CFT correlation functions on the torus, and then evaluate the necessary correlation functions explicitly. Gathering those results in §§3 - 5, we finally evaluate the tachyon amplitude explicitly in §6, where the coupling constants and and of the gauge group are also determined. Finally in §7, we conclude and present some discussions on the relations between the BRS anomaly in the present SFT and the Lorentz invariance anomaly in the light-cone gauge SFT.[9, 10, 11]

For the variables and functions appearing in the one-loop amplitudes, we use the same notations as much as possible as those in Chapter 8 of the textbook by Green, Schwarz and Witten (GSW).[12] We cite in Appendix some modular transformation relations between such one-loop functions which will be used in the text.

## 2 One-loop 2-point tachyon amplitude: preliminaries

The one-loop amplitude obtained by using open 3-string vertex twice contributes to the effective action at one loop as

(9) | |||||

where^{4}^{4}4Note that the factor is multiplied to the
-loop level effective action in the present Feynman rule where the
factors and are omitted from the propagators and the
vertices, respectively.
the open string propagator is given by

(10) | |||||

and the glued vertex is defined by

(11) |

Here is the time interval on the plane defined later and is the anti-ghost factor corresponding to the moduli . Our SFT vertex contains the unoriented projection operators , as an effect of which the propagators of the two intermediate strings and are multiplied by the projection operators:

(12) |

This product of projection operators yields four terms, , and accordingly the glued vertex contains four different configurations as drawn in Fig. 1,

which we call planar (P), nonorientable (or Möbius) (M1 and M2) and nonplanar (NP) diagrams, respectively. To those the following four LPP vertices correspond:

(13) |

where the factor in front of has come from the inner endpoint loop carrying Chan-Paton index in the planar diagram.

We take the external open string states to be on-shell tachyon states

(14) |

Then, the one-loop effective action (9) with Eq. (13) substituted, yields the following one-loop 2-point tachyon amplitudes

(15) |

with an abbreviation , where the individual amplitude is evaluated by the CFT on the corresponding torus P, M1, M2, NP:

(16) |

Here we note two points. First, the RHS is generally multiplied by a factor

(17) |

which is associated with the conformal mapping of the operators from the unit disk to the torus plane and are the positions of punctures on the torus representing the external strings. But here the factor (17) is 1 since the conformal weights are zero for the on-shell tachyons. Secondly, this Eq. (16) determines the CFT correlation functions on the RHS including their signs and weights. This constitutes the content of GGRT;[8, 13] namely, the loop level LPP vertex with P, NO (M1 and M2), NP are already defined above as glued vertices of the two tree level LPP vertices by Eq. (12) with (13). So these torus correlation functions are already fixed including their coefficients. We defer the explicit evaluation of these correlation functions until §5.

Here we first consider the nonplanar diagram NP, for which the two intermediate strings are both twisted. The amplitude corresponding to this diagram alone does not give the full nonplanar amplitude correctly. Indeed, this can easily be seen if we redraw the diagram NP in the form as depicted in the diagram (a) in Fig. 2.

So it can cover only the part of the full nonplanar amplitude, and the remaining part is supplied by the ‘tree’ diagram given by using the open-closed transition vertex twice as drawn in the diagram (b) in Fig. 2.[14] The amplitude for this diagram is given by

(18) | |||||

where the Wick contraction gives the closed string propagator

(19) |

and the following glued LPP vertex has been defined:

(20) |

Again the amplitude is evaluated by referring to the CFT on the torus:

(21) |

The amplitudes in Eq. (15) and in Eq. (18) should smoothly connect with each other at to reproduce the correct nonplanar amplitude, and this condition will determine the coupling constant , as we shall obtain later.

Next consider the two nonorientable diagrams, M1 and M2 terms in Fig. 1. These two diagrams alone do not give the full nonorientable amplitude, again. Indeed, the two diagrams do not connect with each other at the moduli , so that another diagram should exist which interconnects these two configurations at . Such a diagram is just given by the ‘tree’ diagram drawn in Fig. 3 which is obtained by using the vertex.

Clearly the configuration of this diagram coincides at (and ) with that of the M1 diagram at , and at (and ) with that of the M2 diagram at . The amplitude of this diagram is proportional to , and so the smooth connection condition for these amplitudes will determine as we shall see later. The amplitude is given by

(22) | |||||

Finally, we note that the planar diagram in Fig. 1 as and the above diagram of vertex as , both have singularities owing to the closed tachyon and dilaton vanishing into vacuum. As is shown in Fig. 4,

this is almost the same situation as what we have encountered in the disk and amplitudes for closed tachyon 2-point function in the previous paper I. The former planar amplitude is proportional to and the latter amplitude to . The condition for the dilaton contributions to cancel between the two amplitudes will determine of , as we shall explicitly see later.

## 3 Conformal mapping of plane to torus

In order to calculate these amplitudes in Eqs. (15), (18) and (22), we need conformal mapping of the usual unit disk of participating string to the torus for each case. The string world sheets of the ‘light-cone type’ diagrams like Figs. 1, 2 etc, are called plane, on which the complex coordinate is identified with (: a real constant) in each string strip (Re), where is the image of the unit disk of string by a simple (conformal) mapping . Therefore we have only to know the conformal mapping of the plane to the torus for each case.

The conformal mapping of the plane to the torus plane with periods 1 and is generically given by the following (generalized) Mandelstam mapping:[15, 12, 16]

(23) |

where are Jacobi functions with periods and . This satisfies a quasi-periodicity

(24) |

The derivative is truly a doubly periodic function, or elliptic function,[17] which is analytic except for the poles at ():

(25) |

where is the logarithmic derivative of the Jacobi function:

(26) |

corresponds to the image of the external string at (). Two interaction points are given by the zeros of this function:

(27) |

By a general property of elliptic functions,[17] a sum rule holds:

(28) |

We now examine the conformal mappings for the cases of planar, nonorientable, nonplanar, and diagrams, separately, and will find relations which determine the torus moduli , the parameter and interaction points in terms of the string length and the moduli parameters of the plane.

### 3.1 Planar diagram (P)

The mapping for the planar diagram case is drawn in Fig. 5.

In this case the period is purely imaginary and denoted by , and the mapping of this Fig. 5 is given by the above Mandelstam mapping (23) with replaced by . The interaction points in the plane are mapped to in the plane. By using the shift invariance on the torus plane and the sum rule (28), we can parametrize the positions of strings (punctures) and interaction points by two real parameters and as

(29) |

By the help of the periodicity (24), we can determine the parameters and as follows. First, taking , for instance, we have and then and , so that

(30) |

Next, note that the bottom line and the top line with , are mapped to the wavy curves - and - of strings and on the plane in Fig. 5. Therefore we have

(31) |

### 3.2 Nonorientable diagrams (M1 and M2)

The mapping of the nonorientable diagram M1 to torus is drawn in Fig. 6.

In this case the fundamental region of the torus is given as indicated in the same figure Fig. 6, and so the period is now given by . Accordingly, the mapping of this Fig. 6 is given by the above Mandelstam mapping (23) with . We can parametrize the positions of strings (punctures) and interaction points by the same equations as Eq. (29) in this case also.

Similarly to the previous planar case, the periodicity (24) determines the parameters and ; from the period 1 we have the same relation as before,

(34) |

Noting that two points separated by , e.g., the points and , on the plane correspond to those separated by on the plane as seen in Fig. 6, and using the period of , we obtain

(35) | |||||

Equations for and the interaction point take the same form as those for the previous planar case aside from the period:

(36) | |||

(37) |

### 3.3 Nonplanar diagram (NP)

The mapping of the nonplanar diagram NP to torus is drawn in Fig. 7.

The period in this case is , the same as in planar case, and so is the Mandelstam mapping (23) with . However, the strings (punctures) and interaction points are placed differently from the planar case as shown in Fig. 7. So we parametrize their positions by real parameters and as

(38) |

From the period 1 and , we again obtain the same relation as before,

(39) |

Noting that two points separated by , e.g., the points and , on the plane correspond to those separated by on the plane as seen in Fig. 7, and using the period of , we find

(40) | |||||

Equations for and the interaction point become in this case

(42) | |||||

### 3.4 diagram

The diagram obtained by using vertex once is a tree diagram from the SFT viewpoint, but is actually a one-loop diagram from the CFT point of view. The mapping of the diagram to torus is drawn in Fig. 8.

The period in this case is , as is seen from the fundamental region of the torus in Fig. 8. The two interaction points in the fundamental region are

(43) |

and we use the parametrization

(44) |

Since and on the plane correspond to a single point on the plane, we find, using the period of ,

(45) | |||||

Then,