Recurrence relations of higher spin BPST vertex operators
for open strings
Chih-Hao Fu,1,*Jen-Chi Lee,1,†Chung-I Tan,2,‡and Yi Yang1,§ 1Department of Electrophysics, National Chiao-Tung University and Physics Division,
National Center for Theoretical Sciences, Hsinchu 300, Taiwan, Republic of China 2Physics Department, Brown University, Providence, Rhode Island 02912, USA
(Received 9 June 2013; published 19 August 2013)
We calculate higher-spin Brower–Polchinski–Strassler–Tan (BPST) vertex operators for an open bosonic string and express these operators in terms of a Kummer function of the second kind. We derive an infinite number of recurrence relations among BPST vertex operators of different string states. These recurrence relations among BPST vertex operators lead to the recurrence relations among Regge string scattering amplitudes discovered recently.
DOI:10.1103/PhysRevD.88.046004 PACS numbers: 11.25.Db, 11.55.Jy
I. INTRODUCTION
Recently, there has been interest to study Regge regime (RR) string scattering amplitudes [1–6] for higher-spin string states [1,7–10]. One of the motivations was to under-stand their intimate link with the scattering amplitudes in the fixed-angle or Gross regime (GR) [11–15]. In the GR, a saddle-point method was used to calculate string-tree am-plitudes [16–19], and the ratios of scattering amam-plitudes among different string states at each fixed mass level can be extracted and were found to be independent of the scattering energy and scattering angle. Alternatively, these ratios can be rederived algebraically by solving linear relations or GR stringy Ward identities from decoupling of zero-norm states (ZNS) [20–22]. More interestingly, the infinite number of these ratios for the GR can be extracted from RR string scattering amplitudes based on summation algorithms for Stirling number identities [23,24].
In contrast to the GR, an infinite number of recurrence relations among higher-spin RR string scattering ampli-tudes was discovered more recently [1]. Instead of RR stringy Ward identities derived from decoupling of ZNS, the calculation was based on recurrence relations of Kummer functions of the second kind [25]. These recur-rence relations among RR amplitudes were considered to be dual to the linear relations among the GR amplitudes discussed above.
In this paper, we study higher-spin Regge string scattering amplitudes from a Brower–Polchinski– Strassler–Tan (BPST) vertex operator approach. Note that in the original BPST paper [2], the authors calculated the case of closed-string and thus Pomeron vertex operators. Here, for simplicity, we will calculate higher-spin BPST
vertex operators at arbitrary mass levels of an open bosonic string.1The calculation can be easily generalized to the closedstring case. We find that all BPST vertex operators can be expressed in terms of Kummer functions of the second kind. We can then derive an infinite number of recurrence relations among BPST vertex operators of different string states. These recurrence relations among BPST vertex operators lead to the recurrence rela-tions among Regge string scattering amplitudes discovered recently [1].
II. FOUR-TACHYON SCATTERING
We will calculate high-energy open-string scatterings in the Regge regime,
s ! 1; pffiffiffiffiffiffit¼ fixed ðbutpffiffiffiffiffiffitÞ 1Þ; (2.1) where
s ¼ ðk1þ k2Þ2 and t ¼ ðk2þ k3Þ2: (2.2)
Note that the convention for s and t adopted here is differ-ent from the original BPST paper in Ref. [2].
We first review the calculation of tachyon BPST vertex operator [2]. The s t channel of an open-string four-tachyon amplitude can be written as
*[email protected] †[email protected] ‡[email protected] §[email protected]
1Taking advantage of Regge factorization, a Pomeron vertex operator VP was introduced in Ref. [2], which allows one to calculate the coupling between the leading closed-string Regge trajectory with any n-particle external state jW i. In this paper, we only consider 4-point scattering for open strings. As such, we only need to treat the coupling of the leading open-string Reggeon to two-particle states. For brevity, we use here the term ‘‘higher-spin BPST vertex operators’’ collectively for the product of the vertex operator for the leading open-string Reggeon with external two-particle states, one of which has a high spin.
A ¼Z1 0 d! ! k1k2ð1 !Þk2k3 ¼Z1 0 d! ! 2s 2ð1 !Þ2t2: (2.3)
Since s ! 1, the integral is dominated around ! ¼ 1. Making the variable transformation ! ¼ 1 x, the inte-gral is dominated around x ¼ 0, and we obtain
A ¼Z1 0 dx ð1 xÞ 2s 2x22t’ Z dx x22te2sx ¼ 1 t 2 s 2 1þt 2 : (2.4)
Alternatively, the integral in A can be expressed as
A ¼Z d!heik1Xð0Þeik2Xð!Þeik3Xð1Þeik4Xð1Þi: (2.5)
One can calculate the operator product expansion (OPE) in the Regge limit:
eik2XðwÞeik3XðzÞ jw zjk2k3eiðk2þk3ÞXðzÞþik2ðwzÞ@XðzÞþ:
This means
eik2Xð!Þeik3Xð1Þð1!Þk2k3eikXð1Þik2ð1!Þ@Xð1Þþhigherpowerof ð1!Þ; k ¼ k
2þk3: (2.6)
In evaluating Eq. (2.5), one can instead carry out the ! integration first in Eq. (2.6) at the operator level to obtain the BPST vertex operator [2],
VBPST¼ Z d!eik2Xð!Þeik3Xð1Þ Z d!ð1 !Þk2k3eikXð1Þik2ð1!Þ@Xð1Þ ¼Z dxxk2k3eikXð1Þik2x@Xð1Þ ¼ 1 t 2 ½ik2@Xð1Þ1þ t 2eikXð1Þ; (2.7)
which leads to the same amplitude as in Eq. (2.4): A ¼ heik1Xð0ÞV BPSTeik4Xð1Þi ¼ 1 t 2 heik1Xð0Þ½ik 2@Xð1Þ1þ t 2eikXð1Þeik4Xð1Þi ¼ 1 t 2 ðk1k2Þ1þ t 2 1 t 2 s 2 1þt 2 : (2.8)
III. HIGHER-SPIN BPST VERTEX A. A spin-2 state
It was shown [1,7,8] that for the 26-dimensional open bosonic string states of the leading order in energy in the Regge limit at mass level, M22 ¼ 2ðN 1Þ, N ¼
P
n;m;l>0npnþ mqmþ lrl are of the form (we choose the
second state of the four-point function to be the higher-spin string state) jpn; qm; rli ¼ Y n>0 ðT nÞpnY m>0 ðP mÞqmY l>0 ðL lÞrlj0; ki; (3.1) where the polarizations of the second particle with mo-mentum k2 on the scattering plane were defined to be
eP¼M12ðE2; k2; 0Þ ¼Mk22 as the momentum polarization,
eL¼M12ðk2; E2; 0Þ as the longitudinal polarization, and
eT ¼ ð0; 0; 1Þ as the transverse polarization, which lies on
the scattering plane. ¼ diagð1; 1; 1Þ. The three
vec-tors eP, eL, and eTsatisfy the completeness relation ¼
P
;ee, where , ¼ 0, 1, 2 and , ¼ P, L, T
and T
1¼ PeT1, T1L2¼ P;eTeL12etc.
In this section, we first consider a simple case of a spin-2 state P
1P1j0i corresponding to the vertex
ð@XPÞ2eik2Xð!Þ. The four-point amplitude of the spin-2
state with three tachyons can be calculated by using the conventional method: Aðq1¼2Þ¼ Z d!heik1Xð0Þð@XPÞ2eik2Xð!Þeik3Xð1Þeik4Xð1Þi ¼Z d!!k1k2ð1 !Þk2k3 ieP k 1 ! þ ie P k 3 1 ! 2 ¼ ðeP k 1Þ2 1 t 2 s 2 t 21 þ 2ðeP k 1ÞðeP k3Þ 2 t 2 s 2 t 2 ðeP k 3Þ2 3 t 2 s 2 t 2þ1 : (3.2)
The momenta of the four particles on the scattering plane are k1 ¼ ðþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2þ M2 1 q ; p; 0Þ; (3.3) k2 ¼ ðþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2þ M2 2 q ; þp; 0Þ; (3.4) k3¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2þ M2 3 q ; q cos ; q sin ; (3.5) k4¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2þ M2 4 q ; þq cos ; þq sin ; (3.6)
where p j~pj, q j~qj, and k2
i ¼ Mi2. The relevant
kine-matics in the Regge limit are [1,7,8]
eP k 1 ’ s 2M2 ; eP k 3 ’ ~ t 2M2 ¼ t M22 M32 2M2 ; (3.7) eL k 1’ s 2M2 ; eL k3 ’ ~t0 2M2 ¼ tþ M22 M23 2M2 (3.8) and eT k 1 ¼ 0; eT k3 ’ ffiffiffiffiffiffi t p ; (3.9)
where ~t and ~t0are related to t by finite mass square terms ~
t ¼ t M2
2 M23; ~t0¼ t þ M22 M23: (3.10)
By using Eq. (3.7), one can easily see that the three terms in Eq. (3.2) share the same order of energy in the Regge limit. We stress that this key observation on the polarizations for higher-spin states was not discussed in Refs. [2,3].
One can calculate the OPE in the Regge limit:
@XP@XPeik2XðwÞeik3XðzÞ jw zjk2k3 @XðzÞPþ ieP k3 w z 2 eikXðzÞþik2ðwzÞ@XðzÞ: This means @XP@XPeik2Xð!Þeik3Xð1Þ ð1 !Þk2k3 @Xð1ÞP ieP k3 1 ! 2 eikXð1Þik2ð1!Þ@Xð1Þ; k ¼ k 2þ k3: (3.11)
One can carry out the ! integration in Eq. (3.11) at the operator level to obtain the BPST vertex operator:
Vðq1¼2Þ BPST ¼ Z d!ð@XPÞ2eik2Xð!Þeik3Xð1Þ Z d!ð1 !Þk2k3 @Xð1ÞP ieP k3 1 ! 2 eikXð1Þik2ð1!Þ@Xð1Þ ¼ @Xð1ÞP@Xð1ÞPZ dxxk2k3eikXð1Þik2x@Xð1Þ 2ieP k 3@Xð1ÞP Z dxxk2k31eikXð1Þik2x@Xð1Þ ðeP k 3Þ2 Z dxxk2k32eikXð1Þik2x@Xð1Þ ¼ 1 t 2 ½ik2@Xð1Þ t 21@Xð1ÞP@Xð1ÞPeikXð1Þ 2ieP k 3 2 t 2 ½ik2@Xð1Þ t 2@Xð1ÞPeikXð1Þ ðeP k 3Þ2 3 t 2 ½ik2@Xð1Þ t 2þ1eikXð1Þ: (3.12)
We can use this BPST vertex operator to rederive the amplitude
Aðq1¼2Þ ¼ heik1Xð0ÞVðq1¼2Þ BPST eik4Xð1Þi ¼ 1 t 2 heik1Xð0Þ½ik 2@Xð1Þ t 21@Xð1ÞP@Xð1ÞPeikXð1Þeik4Xð1Þi 2ieP k 3 2 t 2 heik1Xð0Þ½ik 2@Xð1Þ t 2@Xð1ÞPeikXð1Þeik4Xð1Þi ðeP k 3Þ2 3 t 2 heik1Xð0Þ½ik 2@Xð1Þ t 2þ1eikXð1Þeik4Xð1Þi eP k 1 2 1 t 2 s 2 t 21 þ 2ðeP k 1ÞðeP k3Þ 2 t 2 s 2 t 2 ðeP k 3Þ2 3 t 2 s 2 t 2þ1 ; (3.13)
which is the same as the amplitude in Eq. (3.2). Note that the three terms in Eq. (3.12) lead to the three terms, respectively, in Eq. (3.13) with the same order of energy in the Regge limit.
B. Higher-spin states We now consider the higher-spin state
jpn; qmi ¼ Y n¼1 ðT nÞpnY m¼1 ðP mÞqmj0i; (3.14)
which corresponds to the vertex
V2ð!Þ ¼ Y n¼1 ð@nXTÞpnY m¼1 ð@mXPÞqm eik2Xð!Þ: (3.15)
The four-point amplitude of the above state with three tachyons was calculated to be (from now on, we set M2¼ M) [1,7,8]
Aðpn;qmÞ¼ Z d!heik1Xð0ÞV 2ð!Þeik3Xð1Þeik4Xð1Þi ¼ 1 M q 1 U q1; t 2þ 2 q1; ~ t 2 B 1 s 2; 1 t 2 Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pn Y m¼2 ~ tðm 1Þ! 1 2M q m (3.16) 1 M q 1 U q1; t 2þ 2 q1; ~ t 2 1 t 2 s 2 1þt 2 (3.17) Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pnY m¼2 ~ tðm 1Þ! 1 2M q m ; (3.18)
where U is the Kummer function of the second kind and is defined to be Uða; c; xÞ ¼ sin c Mða; c; xÞ ða cÞ!ðc 1Þ! x 1cMða þ 1 c; 2 c; xÞ ða 1Þ!ð1 cÞ! ; ðc Þ 2; 3; 4 . . .Þ: (3.19) In Eq. (3.19), Mða; c; xÞ ¼P1j¼0 ðaÞj ðcÞj xj
j! is the Kummer function of the first kind. Here, ðaÞj¼ aða þ 1Þða þ 2Þ . . . ða þ
j 1Þ is the Pochhammer symbol. It is important to note that in Eq. (3.17), c ¼ cðtÞ and is not a constant as in the usual definition, so U in the Regge string scattering amplitudes is not a solution of the Kummer equation.
One can calculate the OPE in the Regge limit,
V2ð!Þeik3Xð1Þ ¼ Y n¼1 ð@nXTÞpnY m¼1 ð@mXPÞqm eik2Xð!Þeik3Xð1Þ Y n¼1 ðn 1Þ!k3 eT ð1 !Þn p nY m¼2 ðm 1Þ!k3 eP ð1 !Þm q m @Xð1Þ eP ik3 eP 1 ! q 1 ð1 !Þk2k3eikXð1Þik2ð1!Þ@Xð1Þ (3.20) ¼~t 2M q 1Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pnY m¼2 ~ tðm 1Þ! 1 2M q m Xq1 j¼0 q1 j ! 2iM2@Xð1Þ eP ~ t j ð1 !Þk2k3NþjeikXð1Þik2ð1!Þ@Xð1Þ; (3.21)
where N ¼Pn;mðnpnþ mqmÞ is the mass level of the higher-spin vertex operator V2ð!Þ. As in the previous calculation,
Vðpn;qmÞ BPST ¼ Z d!V2ð!Þeik3Xð1Þ ~t 2M q 1Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pnY m¼2 ~ tðm 1Þ! 1 2M q m Xq1 j¼0 q1 j ! 2iM2@Xð1Þ eP ~ t j Z d!ð1 !Þk2k3NþjeikXð1Þik2ð1!Þ@Xð1Þ ¼~t 2M q 1Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pnY m¼2 ~tðm 1Þ! 1 2M q m Xq1 j¼0 q1 j ! 2iM@Xð1Þ eP ~ t j Z dxxk2k3NþjeikXð1Þik2x@Xð1Þ ¼~t 2M q 1Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pnY m¼2 ~tðm 1Þ! 1 2M q m Xq1 j¼0 q1 j ! 2iM@Xð1Þ eP ~ t j 1 t 2þ j ½ik2 @Xð1Þ1þ t 2jeikXð1Þ: (3.22)
One notes that, in Eq. (3.22), M@Xð1Þ eP¼ k
2 @Xð1Þ, and the summation over j can be simplified. The BPST vertex
operator can be further reduced to
Vðpn;qmÞ BPST ¼ ~t 2M2 q 1Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pnY m¼2 ~ tðm 1Þ! 1 2M q m Xq1 j¼0 q1j 2 ~ t j 1 t 2 j 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1Þ ¼1 M q 1Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pnY m¼2 ~ tðm 1Þ! 1 2M q m Uq1; t 2þ 2 q1; ~ t 2 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1Þ; (3.23)
where we have used Xl j¼0 l j 2 ~ t j 1 t 2 j ¼ 2lð~tÞlU l;t 2þ 2 l; ~t 2 : (3.24) One notes that the exponent of ½ik2 @Xð1Þ1þ
t 2 in
Eq. (3.23) is mass level N independent. This is related to the fact that the well-known sðtÞ power-law behavior of the four-tachyon string scattering amplitude in the RR can be extended to arbitrary higher-string states and is mass level independent as can be seen from Eq. (3.17). This interesting result was first pointed out in Ref. [7] and will be crucial to derive intermass level recurrence relations among BPST vertex operators to be discussed later.
The BPST vertex operator in Eq. (3.23) leads to exactly the same amplitude as in Eq. (3.18).
IV. RECURRENCE RELATIONS
For any confluent hypergeometric function Uða; c; xÞ with parameters ða; cÞ, the four functions with parameters (a 1, c), (a þ 1, c), (a, c 1), and (a, c þ 1) are called the contiguous functions. A recurrence relation exists
between any such function and any two of its contiguous functions. There are six recurrence relations [25]:
Uða 1; c; xÞ ð2a c þ xÞUða; c; xÞ
þ að1 þ a cÞUða þ 1; c; xÞ ¼ 0; (4.1) ðc a 1ÞUða; c 1; xÞ ðx þ c 1ÞÞUða; c; xÞ
þ xUða; c þ 1; xÞ ¼ 0; (4.2)
Uða; c; xÞ aUða þ 1; c; xÞ Uða; c 1; xÞ ¼ 0; (4.3) ðc aÞUða; c; xÞ þ Uða 1; c; xÞ xUða; c þ 1; xÞ ¼ 0; (4.4)
ða þ xÞUða; c; xÞ xUða; c þ 1; xÞ
þ aðc a 1ÞUða þ 1; c; xÞ ¼ 0; (4.5) ða þ x 1ÞUða; c; xÞ Uða 1; c; xÞ
From any two of these six relations, the remaining four recurrence relations can be deduced.
The confluent hypergeometric function Uða; c; xÞ with parameters (a m, c n) for m, n ¼ 0; 1; 2 . . . are called associated functions. Again, it can be shown that there exist relations between any three associated functions, so that any confluent hypergeometric function can be ex-pressed in terms of any two of its associated functions.
Recently, it was shown [1] that recurrence relations exist among higher-spin Regge string scattering amplitudes of different string states. The key to derive these relations was to use recurrence relations and the addition theorem of Kummer functions. In view of the form of higher-spin BPST vertex operators in Eq. (3.23), one can easily calcu-late recurrence relations among higher-spin BPST vertex operators. By using the recurrence relation of Kummer functions [1], for example,
U 2;t 2; t 2 þt 2þ 1 U 1;t 2; t 2 t 2U 1;t 2þ 1; t 2 ¼ 0; (4.7)
one can obtain the following recurrence relation among BPST vertex operators at mass level M2 ¼ 2:
MpffiffiffiffiffiffitVðq1¼2Þ BPST t2V
ðp1¼1;q1¼1Þ
BPST ¼ 0: (4.8)
Rather than constant coefficients in the RR stringy Ward identities derived in Ref. [1], the coefficients of this recur-rence relation Eq. (4.8) among BPST vertex operators are kinematic variable dependent, similar to BCJ relations among field theory amplitudes [26–30]. The recurrence relation among BPST vertex operators in Eq. (4.8) leads to the recurrence relation among Regge string scattering amplitudes [1]:
MpffiffiffiffiffiffitAðq1¼2Þ t
2A
ðp1¼1;q1¼1Þ¼ 0: (4.9)
V. MORE GENERAL RECURRENCE RELATIONS To derive more general recurrence relations, we need to calculate the BPST vertex operators corresponding to the general higher-spin states in Eq. (3.1). We first calculate the BPST vertex operator corresponding to the state
jpn; rli ¼ Y n¼1 ðT nÞpnY m¼1 ðL lÞrlj0i: (5.1)
The calculation is very similar to that of Eq. (3.14) up to some modification. One can easily get that Eq. (3.22) is now replaced by Vðpn;rlÞ BPST ¼ ~t0 2M r 1Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pn Y l¼2 ~t0ðl 1Þ! 1 2M r l Xr1 j¼0 r1 j ! 2iM@Xð1Þ eL ~ t0 j 1 t 2þ j ½ik2 @Xð1Þ1þ t 2jeikXð1Þ: (5.2)
One notes that, in Eq. (5.2), M@Xð1Þ eLÞ k
2 @Xð1Þ,
and, in contrast to Eq. (3.22), the two factors with expo-nents j and j do not cancel out. The BPST vertex operator for this case thus reduces to
Vðpn;rlÞ BPST ¼ 1 M r 1Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pn Y l¼2 ~t0ðl 1Þ! 1 2M r l Ur1; t 2þ 2 r1; ~ t0 2 eP @Xð1Þ eL @Xð1Þ 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1Þ: (5.3)
The BPST vertex operator in Eq. (5.3) leads to the amplitude Aðpn;rlÞ¼ 1 M r 1 U r1; t 2þ 2 r1; ~ t0 2 1 t 2 s 2 1þt 2 Y n¼1 ½pffiffiffiffiffiffitðn 1Þ!pnY l¼2 ~t0ðl 1Þ! 1 2M r l ; (5.4)
which is consistent with the one calculated in Refs. [1,7,8]. Note that the contribution of eP@Xð1Þ
eL@Xð1Þ in the correlation
function reduces to 1 in the Regge limit by using the first equations of Eqs. (3.7) and (3.8). One sees that Eq. (5.4) can be obtained from Eq. (3.18) by doing the replacement ~
t ! ~t0.
We are now ready to calculate the BPST vertex operator corresponding to the most general Regge state in Eq. (3.1). Similar to the RR amplitude calculated in Ref. [1], the BPST vertex operator can be expressed in two equivalent forms:
Vðpn;qm;rlÞ BPST ¼ Y n¼1 ½ðn1Þ!pffiffiffiffiffiffitpnY m¼1 ðm1Þ! ~t 2M q m Y l¼2 ðl1Þ! ~t0 2M r l 1 M r 1 1 t 2 ½ik2@Xð1Þ1þ t 2eikXð1Þ Xq1 i¼0 q1 i !2 ~ t i t 21 iU r1; t 2þ2ir1; ~ t0 2 eP@Xð1Þ eL@Xð1Þ (5.5) ¼Y n¼1 ½ðn 1Þ!pffiffiffiffiffiffitpn Y m¼2 ðm 1Þ! ~t 2M q m Y l¼1 ðl 1Þ! ~t0 2M r l 1 M q 1 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1Þ Xr1 j¼0 r1 j ! 2 ~ t0 eL @Xð1Þ eP @Xð1Þ j t 2 1 jU q1; t 2þ 2 j q1; ~ t 2 : (5.6)
In the first form, Eq. (5.5), the summationPr1
j¼0has been carried out to produce the Kummer function while, in the second
form, Eq. (5.6), the summationPq1
i¼0has been carried out instead to produce the Kummer function. Either form, Eq. (5.5) or
(5.6), of the above BPST vertex operator leads consistently to the amplitude calculated previously [1]:
Aðpn;qm:rlÞ¼ Y n¼1 ½ðn 1Þ!pffiffiffiffiffiffitpn Y m¼1 ðm 1Þ! ~t 2M q m Y l¼2 ðl 1Þ! ~t0 2M r l 1 M r 1 1 t 2 s 2 1þt 2 (5.7) Xq1 i¼0 q1 i ! 2 ~ t i t 2 1 i U r1; t 2þ 2 i r1; ~ t0 2 (5.8) ¼Y n¼1 ½ðn 1Þ!pffiffiffiffiffiffitpn Y m¼2 ðm 1Þ! ~t 2M q m Y l¼1 ðl 1Þ! ~t0 2M r l 1 M q 1 1 t 2 s 2 1þt 2 Xr1 j¼0 r1 j ! 2 ~ t0 j t 2 1 jU q1; t 2þ 2 j q1; ~ t 2 : (5.9)
Note that, for rl¼ 0, Eq. (5.9) reduces to Eq. (3.18) as expected. One can now derive more general recurrence relations
among BPST vertex operators. As an example, the three BPST vertex operators Vq1¼3 BPST, V
p1¼1;q1¼2 BPST , and V
q1¼2;r1¼1 BPST can be
calculated by using Eq. (5.6) to be
Vðq1¼3Þ BPST ¼ 1 M 3 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1ÞU 3;t 2 1; t 2 1 ; (5.10) Vðp1¼1;q1¼2Þ BPST ¼ 1 M 2 ffiffiffiffiffiffi t p 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1ÞU 2;t 2; t 2 1 ; (5.11) Vðq1¼2;r1¼1Þ BPST ¼ t þ 6 2M 1 M 2 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1Þ U 2;t 2; t 2 1 þ 2 t þ 6 t 2 1 U 2;t 2 1; t 2 1 eL @Xð1Þ eP @Xð1Þ : (5.12)
The recurrence relation among Kummer functions derived from Eq. (4.4) [1],
U 3;t 2 1; t 2 1 þt 2þ 1 U 2;t 2 1; t 2 1 t 2 1 U 2;t 2; t 2 1 ¼ 0; (5.13)
leads to the following recurrence relation among BPST vertex operators at mass level M2 ¼ 4:
MpffiffiffiffiffiffiteL @Xð1ÞVq1¼3 BPSTþ M ffiffiffiffiffiffi t p eP @Xð1ÞVq1¼2;r1¼1 BPST t 2þ 3 eP @Xð1Þ t 2 1 eL @Xð1Þ Vp1¼1;q1¼2 BPST ¼ 0: (5.14)
In addition to the t dependence, the coefficients of the recurrence relation in Eq. (5.14) are operator dependent. The recurrence relation among BPST vertex operators in Eq. (5.14) leads to the recurrence relation among Regge string scattering amplitudes [1]:
MpffiffiffiffiffiffitAðq1¼3Þ 4Aðp1¼1;q1¼2Þþ M ffiffiffiffiffiffit
p
Aðq1¼2;r1¼1Þ ¼ 0:
(5.15)
For the next example, we construct an intermass level recurrence relation for BPST vertex operators at mass level M2 ¼ 2, 4. We begin with the addition theorem of the Kummer function [25], Uða; c; x þ yÞ ¼X 1 k¼0 1 k!ðaÞkð1Þ kykUða þ k; c þ k; xÞ; (5.16)
which terminates to a finite sum for a nonpositive integer a. By taking, for example, a ¼ 1, c ¼t
2þ 1, x ¼2t 1 and
y ¼ 1, the theorem gives [1] U 1;t 2þ 1; t 2 U1;t 2þ 1; t 2 1 U0;t 2þ 2; t 2 1 ¼ 0: (5.17)
Equation (5.17) leads to an intermass level recurrence relation among BPST vertex operators,
Mð2Þðt þ 6ÞVðp1¼1;q1¼1Þ BPST 2Mð4Þ2 ffiffiffiffiffiffi t p Vðq1¼1;r2¼1Þ BPST þ 2Mð4ÞVðp1¼1;r2¼1Þ BPST ¼ 0; (5.18)
where masses Mð2Þ ¼pffiffiffi2, Mð4Þ ¼pffiffiffi4¼ 2, and Vp1¼1;q1¼1
BPST are BPST vertex operators at mass level M2 ¼
2, and Vq1¼1;r2¼1 BPST , V
p1¼1;r2¼1
BPST are BPST vertex operators at
mass levels M2 ¼ 4. In deriving Eq. (5.18), it is important
to use the fact that the exponent of ½ik2 @Xð1Þ1þ t 2in the
BPST vertex operator in Eq. (5.6) is mass level N inde-pendent as mentioned in the paragraph after Eq. (3.24). The recurrence relation among BPST vertex operators in Eq. (5.18) leads to the recurrence relation among Regge string scattering amplitudes [1]:
Mð2Þðt þ 6ÞAðp1¼1;q1¼1Þ 2Mð4Þ2 ffiffiffiffiffiffit
p
Aðq1¼1;r2¼1Þ
þ 2Mð4ÞAðp1¼1;r2¼1Þ¼ 0: (5.19)
In Ref. [1], it was shown that, at each fixed mass level, each Kummer function in the summation of Eq. (5.9) can be expressed in terms of Regge string scattering amplitudes
Aðpn;qm:rlÞat the same mass level. For general values of a,
any Kummer function Uða; c; xÞ can be expressed in terms of two of its associated functions, while for nonpositive integer values of a in the RR string amplitude case, one can further fixes Uða; c; xÞ up to an overall factor by using Kummer function recurrence relations [1]. As a result, all Regge string scattering amplitudes can be algebraically solved by Kummer function recurrence relations up to multiplicative factors. An important application of the above properties is the construction of an infinite number of recurrence relations among Regge string scattering am-plitudes. One can use the recurrence relations of Kummer functions Eqs. (4.1) to (4.6) to systematically construct recurrence relations among Regge string scattering amplitudes.
In view of the form of BPST vertex operators calculated in Eq. (5.6), one can similarly solve [1] all Kummer functions Uða; c; xÞ in Eq. (5.6) in terms of BPST vertex operators and use the recurrence relations of Kummer functions Eqs. (4.1) to (4.6) to systematically construct an infinite number of recurrence relations among BPST vertex operators. Moreover, the forms of all BPST vertex operators can be fixed by these recurrence relations up to multiplicative factors. These recurrence relations among BPST vertex operators are dual to linear relations or sym-metries among high-energy fixed-angle string scattering amplitudes discovered previously [16–19].
We illustrate the prescription here to construct other examples of recurrence relations among BPST vertex op-erators at mass level M2¼ 4. Generalization to arbitrary
mass levels will be given in the next section. There are 22 BPST vertex operators for the mass level M2¼ 4. We first
consider the group of BPST vertex operators with q1 ¼ 0,
ðVTTT
BPST; VBPSTLTT; VBPSTLLT; VBPSTLLLÞ [1]. The corresponding r1
for each BPST vertex operator is (0, 1, 2, 3). Here, we use a new notation for the BPST vertex operator, for example, VLLT BPST V ðp1¼1;r1¼2Þ BPST , VBPSTLT ¼ V ðp1¼1;r2¼1Þ BPST , VBPSTTL ¼ V ðp2¼1;r1¼1Þ
BPST , etc. By using Eq. (5.6), one can easily
calculate that VBPSTTTT ¼ ð ffiffiffiffiffiffi t p Þ3 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1Þ U0;t 2þ 2; t 2 1 ; (5.20) VLTT BPST¼ t þ 6 2M ð ffiffiffiffiffiffi t p Þ2 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1Þ U0;t 2þ 2; t 2 1 þ 2 t þ 6 t 2 1 U0;t 2þ 1; t 2 1 eL @Xð1Þ eP @Xð1Þ ; (5.21)
VLLT BPST¼ t þ 6 2M 2 ðpffiffiffiffiffiffitÞ1 t 2 ½ik2@Xð1Þ1þ t 2eikXð1Þ U 0;t 2þ2; t 2 1 þ 4 t þ 6 t 2 1 U 0;t 2þ 1; t 21 eL@Xð1Þ eP@Xð1Þþ 2 t þ 6 2 t 21 t 2 U 0;t 2; t 21 eL@Xð1Þ eP@Xð1Þ 2 ; (5.22) VBPSTLLL ¼ t þ 6 2M 3 1 t 2 ½ik2 @Xð1Þ1þ t 2eikXð1Þ U 0;t 2þ 2; t 2 1 þ 6 t þ 6 t 2 1 U 0;t 2þ 1; t 2 1 eL @Xð1Þ eP @Xð1Þ þ 3 2 t þ 6 2 t 2 1 t 2 U 0;t 2; t 2 1 eL @Xð1Þ eP @Xð1Þ 2 þ 2 t þ 6 3 t 2 1 t 2 t 2þ 1 U 0;t 2 1; t 2 1 eL @Xð1Þ eP @Xð1Þ 3 : (5.23)
From the above equations, one can easily see that Uð0;2tþ 2;2t 1Þ can be expressed in terms of VBPSTTTT , Uð0;2tþ 1;2t 1Þ
can be expressed in terms of (VBPSTTTT, VBPSTLTT ), Uð0;2t;2t 1Þ can be expressed in terms of (VBPSTTTT , VBPSTLTT, VBPSTLLT), and finally
Uð0;2t 1;2t 1Þ can be expressed in terms of (VBPSTTTT , VBPSTLTT , VBPSTLLT, VBPSTLLL). So all Kummer functions can be solved and
expressed in terms of BPST vertex operators. We have
Uð0;t 2þ 2; t 2 1Þ ¼ 1ðpffiffiffiffiffiffitÞ3 VBPSTTTT ; (5.24) Uð0;t 2þ 1; t 2 1Þ ¼ 1ðpffiffiffiffiffiffitÞ3t þ 6 t þ 2 eP @Xð1Þ eL @Xð1Þ VTTT BPST 2M t þ 6 ffiffiffiffiffiffi t p VLTT BPST ; U 0;t 2; t 2 1 ¼ 1ðpffiffiffiffiffiffitÞ3ðt þ 6Þ2 tðt þ 2Þ eP @Xð1Þ eL @Xð1Þ 2 VTTT BPST 2 2M t þ 6 ffiffiffiffiffiffi t p VLTT BPSTþ 2M t þ 6 ffiffiffiffiffiffi t p 2 VLLT BPST ; (5.25) U 0;t 2 1; t 2 1 ¼ 1ðpffiffiffiffiffiffitÞ3 ðt þ 6Þ3 tðt2 4Þ eP @Xð1Þ eL @Xð1Þ 3 VBPSTTTT 3 2M t þ 6 ffiffiffiffiffiffi t p VBPSTLTT þ 3 2M t þ 6 ffiffiffiffiffiffi t p 2 VLLT BPST 2M t þ 6 ffiffiffiffiffiffi t p 3 VLLL BPST ; (5.27) where ð1 2tÞ½ik2 @Xð1Þ1þ t
2eikXð1Þ. To derive an example of the recurrence relation, one notes that Eq. (4.2)
gives t 2U 0;t 2; t 21 ðt1ÞU0;t 2þ1; t 21 þt 21 U 0;t 2þ2; t 21 ¼ 0; (5.28)
which leads to the recurrence relation among BPST vertex operators: t 2 1 ðt 1Þðt þ 6Þ t þ 2 eP @Xð1Þ eL @Xð1Þþ ðt þ 6Þ2 2ðt þ 2Þ eP @Xð1Þ eL @Xð1Þ 2 VTTT BPST þðt 1Þ t þ 2 eP @Xð1Þ eL @Xð1Þ ðt þ 6Þ ðt þ 2Þ eP @Xð1Þ eL @Xð1Þ 2 ð2MpffiffiffiffiffiffitÞVLTT BPSTþ 1 2ðt þ 2Þ eP @Xð1Þ eL @Xð1Þ 2 ð2MpffiffiffiffiffiffitÞ2 VBPSTLLT ¼ 0: (5.29) Again, one can use Eq. (5.29) to deduce the recurrence relation among Regge string scattering amplitudes:
ðt þ 22ÞAðp1¼3Þ 14M ffiffiffiffiffiffit
p
Aðp1¼2;r1¼1Þþ 2M2ð ffiffiffiffiffiffit
p
Þ2Aðp1¼1;r1¼2Þ¼ 0: (5.30)
Other recurrence relations of Kummer functions can be used to derive more recurrence relations among BPST vertex operators. For example, Eq. (4.2) gives a recurrence relation of Uð0;2tþ 1;2t 1Þ and its associated functions
Uð0;2t 1;2t 1Þ and Uð0;2tþ 2;2t 1Þ
tU 0;t 2 1; t 2 1 ð3t 4ÞU0;t 2þ 1; t 2 1 þ 2ðt 2ÞU0;t 2þ 2; t 2 1 ¼ 0; (5.31)
2ðt 2Þ ð3t 4Þðt þ 6Þ t þ 2 eP @Xð1Þ eL @Xð1Þþ ðt þ 6Þ3 ðt2 4Þ eP @Xð1Þ eL @Xð1Þ 3 VTTT BPSTþ ð3t 4Þ t þ 2 eP @Xð1Þ eL @Xð1Þ 3ðt þ 6Þ2 ðt2 4Þ eP @Xð1Þ eL @Xð1Þ 3 ð2MpffiffiffiffiffiffitÞVLTT BPSTþ 3ðt þ 6Þ ðt2 4Þ eP @Xð1Þ eL @Xð1Þ 3 ð2MpffiffiffiffiffiffitÞ2VLLT BPST 1 ðt2 4Þ eP @Xð1Þ eL @Xð1Þ 3 ð2MpffiffiffiffiffiffitÞ3VLLL BPST¼ 0: (5.32)
One can use Eq. (5.32) to deduce the recurrence relation among Regge string scattering amplitudes:
ð3t2þ 76t þ 92ÞAðp1¼3Þ 2ð23t þ 50ÞM ffiffiffiffiffiffit p Aðp1¼2;r1¼1Þþ 6M2ðt þ 6Þð ffiffiffiffiffiffit p Þ2Aðp1¼1;r1¼2Þ 4M3ð ffiffiffiffiffiffit p Þ3Aðr1¼3Þ¼ 0: (5.33)
Similarly, we can consider groups of BPST vertex opera-tors ðVPT
BPST; VBPSTPL Þ, ðVBPSTLT ; VBPSTLL Þ, and ðVBPSTTT ; VBPSTTL Þ
with q1¼ 0; a group of BPST vertex operators
ðVPTT
BPST; VBPSTPLT; VBPSTPLLÞ with q1 ¼ 1; and group of BPST
vertex operators ðVBPSTPPT; VBPSTPPLÞ with q1 ¼ 2. All the
re-maining 7 BPST vertex operators are with r1 ¼ 0, and each
BPST vertex operator contains only one Kummer function. Thus, all Kummer functions involved at mass level M2¼ 4
can be algebraically solved and expressed in terms of BPST vertex operators. One can then use recurrence rela-tions of Kummer funcrela-tions to derive more recurrence relations among the BPST vertex operators.
VI. ARBITRARY MASS LEVELS
In this section, we solve the Kummer functions in terms of the highest-spin string states scattering amplitudes for arbitrary mass levels. The highest-spin string states at the mass level M2 ¼ 2ðN 1Þ are defined as
jN q1 r1; q1; r1i ¼ ðT1ÞNq1r1ðP1Þq1
ðL
1Þr1j0; ki; (6.1)
where only the 1 operator appears. The highest-spin
string states BPST vertex operators can be easily obtained from Eq. (5.6) as ðVTÞNq1r1ðVPÞq1ðVLÞr1 VðNq1r1;q1;r1Þ BPST ¼ t 2 1 ½ik2 @Xð1Þ1þ t 2eikXð1Þð ffiffiffiffiffiffit p ÞNq1r1 1 M q 1 ~t0 2M r 1 Xr1 j¼0 r1 j !2 ~ t0 eL @Xð1Þ eP @Xð1Þ j t 2 1 jU q1; t 2þ 2 j q1; ~ t 2 : (6.2) In view of the form of Eq. (5.27), we can solve the Kummer function from Eq. (6.2) and express it in terms of the highest-spin BPST vertex operators as
U q1; t 2þ 2 q1 r1; ~ t 2 ¼ð2t 1Þ ðt 2 1Þr1 ½ik2 @Xð1Þ1þ t 2eikXð1Þ ðMVPÞq1 VT ffiffiffiffiffiffi t p Nq1 eP @Xð1Þ eL @Xð1Þ ffiffiffiffiffiffi t p MV L VT ~ t0 2 r 1 : (6.3)
Putting the Kummer functions (6.3) into the recurrence relations (4.1), (4.2), (4.3), (4.4), (4.5), and (4.6), we can then obtain recurrence relations among BPST vertex operators.
Let us consider, for example, the recurrence relation
ðc a 1ÞUða; c 1; xÞ ðx þ c 1ÞÞUða; c; xÞ þ xUða; c þ 1; xÞ ¼ 0: (6.4) With a ¼ q1; c ¼ t 2þ 1 q1 r1; x ¼ ~ t 2¼ t M2þ 2 2 ; (6.5)
t 2 r1 U q1; t 2 q1 r1; ~ t 2 ~t 2þ t2 q1 r1 U q1; t 2þ 1 q1 r1; ~ t 2 þ~t 2U q1; t 2þ 2 q1 r1; ~ t 2 ¼ 0: (6.6)
Plugging the Kummer functions (6.3) into the above re-currence relation, we obtain the rere-currence relation among BPST vertex operators at general mass level N,
ðVPÞq1ðVTÞNq1ðXÞr1 X2þ ~ t 2þ t2 q1 r1 X þ~t 2 t 2þ 1 r1 ¼ 0; (6.7)
where we have defined
X e P @Xð1Þ eL @Xð1Þ ffiffiffiffiffiffi t p MV L VT ~ t0 2 ¼ eP @Xð1Þ eL @Xð1Þ ffiffiffiffiffiffi t p MV L VT t þ M2þ 2 2 : (6.8) As an example, at the mass level M2¼ 4 with q
1 ¼ r1¼ 0, we get ðVTÞ3 X2þ ðt 1ÞX þ t2 4 1 ¼ 0; (6.9) where X ¼e P @Xð1Þ eL @Xð1Þ ffiffiffiffiffiffi t p MV L VT t þ 6 2 : (6.10)
A simple calculation shows that Eq. (6.9) is exactly the same as Eq. ((5.29)), and the same recurrence relation among Regge string scattering amplitudes ((5.30)) follows.
VII. DISCUSSION
Although we focus here on the spin dependence of the four-point open-string amplitudes, it is useful to briefly recall the generality of the BPST vertex operator, which emphasizes Regge factorization and can be applied to arbitrary n-point amplitudes, n 4. A Regge limit is defined by singling out a longitudinal direction, e.g., the z axis, along which all momenta are large while keeping transverse components, p?, fixed. We separate particles
into two groups, the right-moving and left-moving, with large pþand plarge, respectively. Each can have nRand
nLstates, with nRþ nL¼ n and nR, nL 2. Within each
group, relative momenta remain finite in the Regge limit. Any n-point open-string amplitude can formally be ex-pressed in a factorable form AL;R¼
R
dwhWRwL02WLi,
where WRand WLare products of respective right-moving
and left-moving vertex operators, with all world sheet integrations done except one, i.e., w. The last remaining integration is such that the factor wL0 corresponds to
over-all rescaling in the world sheet coordinates in WL. (For
more details, see Ref. [2].) In the Regge limit, the ampli-tude AL;Rtakes on a simply factorized form, and it can be
expressed in terms of the BPST vertex operator, AL;R¼ hWRViðtÞhVþWLi
¼ hWR;0VifðtÞsðtÞghVþWL;0i; (7.1)
where ðtÞ is the leading Regge trajectory, with 0¼ 1=2, and ðtÞ is a Regge propagator, given by a Gamma func-tion. Here, V are BPST vertex operators, which are ‘‘on shell’’ along the leading trajectory. This is the most general form of Regge factorization for any number of external particles. The factors hWR;0Vi and hVþWL;0i are
general-ized (nRþ 1)- and (nLþ 1)-point on-shell amplitudes,
evaluated in the respective rest frame, with one external line being on the leading Regge trajectory. Each, due to Mobius invariance, involves nR 2 and nL 2 world
sheet integrations.
We have studied in this paper the Regge behavior of four-point open-string scattering amplitudes, with one par-ticle having arbitrary high spin and three others being tachyons, using the technique of the BPST vertex operator. Since we only work with four-point amplitudes in this paper, nR¼ nL¼ 2, there is no integration involved for
hWR;0Vi and hVþWL;0i, due to Mobius invariance. In
particular, WL involves two tachyons. Since one can
show that hVþWL;0i is simply a constant, therefore, what
we have calculated is simply hWR;0Vi, with WRa product
of two vertex operators, one for a tachyon and another for a string state with arbitrary spin. For brevity, we have col-lectively referred to WR;0Vas BPST vertex operators. The
generalization of our analysis to amplitudes for n ¼ 5; 6 . . . will be treated elsewhere.
We have derived in this paper an infinite number of recurrence relations among these matrix elements of the BPST vertex operator between different string states with different spins, which can be expressed in terms of a Kummer function of the second kind. These recurrence relations lead to the same recurrence relations among Regge string scattering amplitudes recently discovered in Ref. [1] by a more traditional method. We show that all Kummer functions involved at each fixed mass level can be algebraically solved and expressed in terms of BPST ver-tex operators. We give a prescription to construct recur-rence relations among BPST vertex operators. For illustration, we calculate some examples of recurrence relations among BPST vertex operators of different string states based on recurrence relations of Kummer functions together with the addition theorem of Kummer function. We stress that, although the higher-spin BPST vertex op-erators were considered in Refs. [2,3], the key observation on the energy orders in the Regge limit from polarizations of higher-spin states was not discussed in Refs. [2,3]. One cannot obtain recurrence relations among higher-spin BPST vertex operators in the Regge limit without includ-ing the energy orders from these higher-spin polarizations.
The recurrence relations among BPST vertex operators lead to the recurrence relations among Regge string scat-tering amplitudes. They are thus both closely related to Regge stringy Ward identities [1] derived from the decou-pling of Regge ZNS in the string spectrum. These recur-rence relations are dual to linear relations derived from ZNS or symmetries among high-energy fixed-angle string scattering amplitudes [16–19].
ACKNOWLEDGMENTS
We thank Marko Djuric, Song He, Yu-Ting Huang, and Yoshihiro Mitsuka for helpful discussions. This work is supported in part by the National Science Council, 50 Billions Project of Ministry of Education, National Center for Theoretical Sciences and S. T. Yau Center of NCTU, Taiwan.
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