Decoupling of degenerate positive-norm states in Witten's string field theory

Decoupling of degenerate positive-norm states in Witten’s string field theory

Hsien-Chung Kao*

Department of Physics, Tamkang University, Tamsui, Taiwan, Republic of China

and Department of Physics, National Taiwan Normal University, Taipei, Taiwan, Republic of China Jen-Chi Lee†

Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China 共Received 17 December 2002; published 21 April 2003兲

We show that the degenerate positive-norm physical propagating fields of the open bosonic string can be gauged to the higher rank fields at the same mass level. As a result, their scattering amplitudes can be determined from those of the higher spin fields. This phenomenon arises from the existence of two types of zero-norm states with the same Young representations as those of the degenerate positive-norm states in the old covariant first quantized共OCFQ兲 spectrum. This is demonstrated by using the lowest order gauge transforma-tion of Witten’s string field theory 共WSFT兲 up to the fourth massive level 共spin-five兲, and is found to be consistent with conformal field theory calculation based on the first quantized generalized sigma-model ap-proach. In particular, on-shell conditions of zero-norm states in the OCFQ stringy gauge transformation are found to correspond, in a one-to-one manner, to the background ghost fields in off-shell gauge transformation of WSFT. The implication of decoupling of scalar modes on Sen’s conjectures is also briefly discussed. DOI: 10.1103/PhysRevD.67.086003 PACS number共s兲: 11.25.Sq, 11.25.Db, 11.25.Hf

I. INTRODUCTION

It was pointed out more than ten years ago by Gross 关1兴 that, in addition to the strong coupling regime, the most im-portant nonperturbative regime of string theory is the high-energy stringy (␣

⬘

→⬁) behavior of the theory. It is in this regime that the theory becomes very different from point particle field theory. Among many interesting stringy behav-iors, it was believed that an infinite broken gauge symmetry gets restored at an energy much higher than the Planck en-ergy. Moreover, this symmetry is powerful enough to link different string scattering amplitudes and, in principle, can be used to express all string amplitudes in terms of, say, the dilaton amplitude.

Instead of studying stringy scattering amplitudes关2兴, one alternative to explicitly derive the stringy symmetry is to use the generalized worldsheet sigma-model approach. In this approach, one uses conformal field theory to calculate the equations of motion for massive string background fields in the lowest order weak field approximation, but valid to all orders in ␣

⬘

. The weak field approximation is thus the ap-propriate approximation scheme to study high-energy sym-metry of the string. An infinite set of on-shell stringy gauge symmetry is then derived by requiring the decoupling of both types of zero-norm physical states in the old covariant first quantized 共OCFQ兲 spectrum 关3兴. In particular, all physical propagating states at each fixed mass level are found to form a large gauge multiplet. This begins to show up at the second massive level 共spin-three兲. Moreover, it was remarkable to discover that 关4兴 the degenerate positive-norm physical propagating fields of the third massive level of the open bosonic string can be gauged to the higher rank fields by the

existence of zero-norm states with the same Young represen-tations. It was also shown 关5兴 that the scattering amplitudes of these degenerate positive-norm states can be expressed in terms of those of higher spin states at the same mass level through massive Ward identities. The subtlety of the scalar state pointed out in Ref. 关5兴 will be resolved at the end of Sec. II. This phenomenon begins to show up at the third massive level 共spin-four兲 and was argued to be a sigma-model of n⫹1 loop results for the nth massive level. These stringy phenomena seem to be closely related to the results in Ref. 关1兴. In fact, an infinite number of linear relations between the string tree-level scattering amplitudes of differ-ent string states, similar to those claimed in Ref.关1兴, can be derived by making use of an infinite number of zero-norm states 关5兴. To claim that the decoupling phenomenon persist for general higher levels, it would be very important a priori to see whether one can rederive it from the second quantized off-shell Witten string field theory 共WSFT兲 关6兴.

Recently there is a revived interest in WSFT, mainly due to Sen’s conjecture on tachyon condensation on D-brane关7兴. It becomes more and more clear that a second quantized field theory of string is unavoidable, especially when one wants to study higher string modes. Thus, a cross check by both first and second quantized approaches of any reliable string theory result would be of great importance. Unfortunately, most of the recent researches on string field theory were confined to the scalar modes on identification of nonpertur-bative string vacuum关8兴. Our aim in this paper is to consider the gauge transformation of all string modes with any spin and in arbitrary gauge关9兴. We will first prove the decoupling phenomenon of the third massive level of open bosonic string claimed in Ref.关4兴 by WSFT. The result is then gen-eralized to the fourth massive level by both the first and the second quantized approaches. This paper is organized as fol-lows. In Sec. II we first summarize the previous results ob-tained in the first quantized approach. In Sec. III we explic-*Email address: [email protected]

itly calculate the lowest order gauge transformation level by level up to the third massive level in WSFT, and compare them with those of the first quantized approach. Some im-portant observations will be made for ghost fields in WSFT and zero-norm states in the OCFQ spectrum. The transfor-mation will be separated into the matter and the ghost field parts in WSFT. The matter part is found to be consistent with the previous calculation关5兴 based on the old covariant string field gauge transformation of Banks and Peskin 关10兴. The ghost part is argued to correspond to the lifting of on-shell 共including on-mass-shell, gauge, and traceless兲 conditions of zero-norm states in the OCFQ calculation. Section IV is de-voted to the fourth massive level. Both first and second quan-tized calculations are new and will be presented in Sec. IV. A brief conclusion is made in Sec. V. The lengthy gauge trans-formation of ghost fields for level four will be collected in the Appendix.

II. OLD COVARIANT FIRST QUANTIZED APPROACH

The old covariant quantization is one of the three standard quantization schemes of string. In addition to the physical positive-norm propagating modes, there exist two types of physical zero-norm states in the bosonic open string spec-trum 关11兴. They are as follows:

Type I: L_⫺1兩␹

典

, where Lm兩␹

典

⫽0, m⭓1, L0兩␹

典

⫽0, 共1兲 Type II: 共L_⫺2⫹3 2L⫺1 2 _兲兩␹˜

_典

_, where L_m兩␹˜

典⫽0, m⭓1, 共L

₀⫹1兲兩␹˜

典

⫽0. 共2兲 While type I states have zero-norm at any spacetime dimen-sion, type II states have zero-norm only at D⫽26. Their existence turns out to be important in the following discus-sion. The explicit forms of these zero-norm states have been calculated and their Young tabulation, together with positive-norm states, up to the third massive level, are listed in Table I. Note that zero-norm states are not included in the light-cone quantization.

It was demonstrated in the first order weak field approxi-mation that for each zero-norm state there corresponds an on-shell gauge transformation for the positive-norm back-ground field (␣

⬘⬅

12) 关3兴: m2⫽0, ␦A_␮⫽⳵␮␪, 共3a兲 ⳵2_{␪⫽0; 共3b兲} m2⫽2, ␦B_␮␯⫽⳵(␮␪␯), 共4a兲 ⳵␮_␪ ␮⫽0, 共⳵2⫺2兲␪␮⫽0; 共4b兲 ␦B_␮␯⫽3 2⳵␮⳵␯␪⫺ 1 2␩␮␯␪, 共5a兲 共⳵2_{⫺2兲␪⫽0; 共5b兲} m2⫽4, ␦C_␮␯␭⫽⳵(␮␪_␯␭), 共6a兲 ⳵␮_␪ ␮␯⫽␪␮␮⫽0, 共⳵2⫺4兲␪␮␯⫽0; 共6b兲 ␦C_␮␯␭⫽52⳵(␮⳵␯␪␭) 1 _⫺␩( ␮␯␪␭)1 , 共7a兲 ⳵␮_␪ ␮ 1 ⫽0, 共⳵2_⫺4兲␪ ␮ 1_{⫽0; 共7b兲} ␦C_␮␯␭⫽1 2⳵(␮⳵␯␪␭) 2 _{⫺2␩(␮␯␪} ␭) 2 _, ␦C[␮␯]⫽9⳵[␮␪_␯] 2 , 共8a兲 ⳵␮_␪ ␮ 2_{⫽0, 共⳵}2_⫺4兲␪ ␮ 2_{⫽0; 共8b兲} ␦C_␮␯␭⫽35⳵␮⳵␯⳵␭␪⫺ 1 5␩(␮␯⳵␭)␪, 共9a兲 共⳵2_{⫺4兲␪⫽0. 共9b兲}

These symmetry transformations can be explicitly shown to be symmetries of the equation of motion for massive back-ground fields, which were calculated in Ref.关3兴. A complete 2D sigma-model renormalization group analysis for the first massive level was done in Refs.关12,13兴. It was noted that the traceless condition of the equation of motion, which was included in Refs. 关3兴, can only be obtained by requiring quantum Weyl invariance at a linearized two-loop approxi-mation. The symmetries considered in Ref. 关12,13兴 corre-sponding to adding worldsheet total derivative terms to the effective Lagrangian, however, turn out to be only the subset of symmetries calculated in Ref. 关3兴. The complete set of symmetries generated by two types of zero-norm states con-sidered in this section include some nontotal derivative terms, e.g., Eq.共5兲.

In the above equations, A, B, C are positive-norm back-ground fields, ␪’s represent zero-norm background fields, and⳵2⬅⳵␮⳵␮. There are on-mass-shell, gauge, and traceless conditions on the transformation parameters ␪’s, which will correspond to Becchi-Rouet-Stora-Tyutin 共BRST兲 ghost fields in a one-to-one manner in WSFT, as will be discussed in the next section. Equation 共3兲 is of course the usual on-shell gauge transformation, and Eq. 共5兲 is the first residual stringy gauge symmetry. Note that␪_␮1 and␪_␮2 in Eqs.共7兲 and 共8兲 are some linear combination of the original type I and type II vector zero-norm states calculated by Eqs.共1兲 and 共2兲. It is interesting to see that Eq.共8兲 implies that the two second massive level modes C_␮␯␭ and C[␮␯] form a larger gauge multiplet 关3兴. This is a generic feature for higher massive TABLE I. The OCFQ spectrum of open bosonic string.

level and had also been justified from S matrix point of view 关14兴. One might want to generalize the calculation to the second order weak field to see the intermass level symmetry. This however suffers from the so-called nonperturbative nonrenormalizability of 2D ␴ model, and one is forced to introduce infinite number of counter terms to preserve the worldsheet conformal invariance 关15兴.

Instead of calculating the stringy gauge symmetry at level

m2⫽6, we will only concentrate on the equation of motion.

It was discovered that an even more interesting phenomenon begins to show up at this mass level. Take the energy-momentum tensor on the worldsheet boundary in the first order weak field approximation to be of the following form:

T共␶兲⫽⫺1

2␩␮␯⳵␶X␮⳵␶X␯⫹D␮␯␣␤⳵␶X␮⳵␶X␯⳵␶X␣⳵␶X␤ ⫹D␮␯␣⳵␶X␮⳵␶X␯⳵␶2X␣⫹D␮␯0 ⳵␶2X␮⳵␶2X␯

⫹D␮␯1 ⳵␶X␮⳵␶3X␯⫹D␮⳵␶4X␮, 共10兲 where␶is worldsheet time, X⬅X(␶). This is the most gen-eral worldsheet coupling in the gengen-eralized ␴ model ap-proach consistent with vertex operator consideration 关16兴. The conditions to cancel all q number worldsheet conformal anomalous terms correspond to cancelling all kinds of loop divergences 关13兴 up to the four loop orders in the 2D con-formal field theory. It is easier to use T•T operator-product calculation and the conditions read关4兴

2⳵␮D_␮␯␣␤⫺D(␯␣␤)⫽0, 共11a兲 ⳵␮_D ␮␯␣⫺2D␯␣0 ⫺3D␯␣1 ⫽0, 共11b兲 ⳵␮_D ␮␯ 1 _⫺12D ␯⫽0, 共11c兲 3D_␮␯␣␮⫹⳵␮D_␯␣␮⫺3D(␯␣) 1 _{⫽0, 共11d兲} D_␮␯␮⫹4⳵␮D_␮␯0 ⫺24D_␯⫽0, 共11e兲 2D_␮␯␯⫹3⳵␯D_␮␯1 ⫺12D_␮⫽0, 共11f兲 2D_␮0␮⫹3D_␮1␮⫹12⳵␮D_␮⫽0, 共11g兲 共⳵2_{⫺6兲␾⫽0. 共11h兲} Here, ␾ represents all background fields introduced in Eq. 共10兲. It is now clear through Eqs. 共11b兲 and 共11d兲 that both

D_␮␯0 and D₍1_␮␯) can be expressed in terms of D_␮␯␣␤ and

D_␮␯␣. D_[1_␮␯] can be expressed in terms of D_␮␯␣␤and D_␮␯␣ by Eq. 共11b兲. Equations 共11a兲 and 共11c兲 imply that D(␮␯␣) and D_␮can also be expressed in terms of D_␮␯␣␤and mixed-symmetric D_␮␯␣. Finally, Eqs. 共11e兲–共11g兲 are the gauge conditions for D_␮␯␣␤and mixed-symmetric D_␮␯␣ after sub-stituting D_␮␯0 , D_␮␯1 , and D_␮ in terms of D_␮␯␣␤ and mixed-symmetric D_␮␯␣. The remaining scalar particle has auto-matically been gauged to higher rank fields since Eq.共10兲 is

already the most general form of the background-field cou-pling. This means that the degenerate spin-two and scalar positive-norm states can be gauged to the higher rank fields

D_␮␯␣␤ and mixed-symmetric D_␮␯␣ in the first order weak field approximation. In fact, for instance, it can be explicitly shown 关5兴 that the scattering amplitude involving the positive-norm spin-two state can be expressed in terms of those of spin-four and mixed-symmetric spin-three states due to the existence of a type I and a type II spin-two zero-norm states. The subtlety of the scalar state scattering amplitude pointed out in Ref.关5兴 can be resolved in the following way. Take a representative of the scalar state to be 关16兴

:

冋

⫺

冉

␩_␮␯⫹13 3 k␮k␯

冊

⳵z 2 X␮ ⳵_z2X␯ ⫺i

冉

20 9 k␮k␯k␳⫹ 2 3k␮␩␯␳⫹ 13 3 k␳␩␮␯

冊

⳵zX ␮_⳵ zX␯ ⳵z 2_X␳ ⫹

冉

23₈₁k_␮k_␯k_␳k_␴⫹32 27k␮k␯␩␳␴⫹ 19 18␩␮␯␩␳␴

冊

⫻⳵zX␮⳵zX␯ ⳵zX␳ ⳵zX␴

册

eikX(z): .

It turns out that one cannot gauge away the first term in the above equation by using the two scalar zero-norm states. However, we have already known the amplitude correspond-ing to ⳵_z2X␮ ⳵_z2X␯ are fixed by those of the spin-four and mixed-symmetric spin-three states. The totally symmetric spin-three amplitude corresponding to the totally symmetric spin-three part of the second term, ⳵zX(␮ ⳵zX␯ ⳵z

2

X␳), can be fixed by the spin-four amplitude due to the existence of the totally symmetric spin-three zero-norm state. As a result, the scalar state scattering amplitude is again fixed by the amplitudes of spin-four and mixed-symmetric spin-three states. Although all the four-point amplitudes considered in Ref. 关5兴 contain three tachyons, the argument can be easily generalized to more general amplitudes. This is very differ-ent from the analysis of lower massive levels where all positive-norm states have independent scattering amplitudes. Presumably, this decoupling phenomenon comes from the ambiguity in defining positive-norm states due to the exis-tence of zero-norm states in the same Young representations. We will justify this decoupling by WSFT in the next section. Finally, one expects this decoupling to persist even if one includes the higher order corrections in the weak field ap-proximation, as there will be even stronger relations between the background fields order by order through iteration.

III. WITTEN’S STRING FIELD THEORY APPROACH

It would be much more convincing if one can rederive the stringy phenomena discussed in the previous section from WSFT. Not only can one compare the first quantized string with the second quantized string, but also the old covariant quantized string with the BRST quantized string. Although the calculation is lengthy, the results, as we shall see, are still controllable by utilizing the results from first quantized

ap-proach in Sec. II. There exist important consistency checks of first quantized string results from WSFT in the literature, e.g., the rederivation of Veneziano and Kubo-Nielson ampli-tudes from WSFT 关17兴. In some stringy cases, calculations can only be done in the string field theory approach. For example, the recently developed p p wave string amplitudes can only be calculated in the light-cone string field theory 关18兴. Sen’s recent conjectures of tachyon condensation on D-brane again were mostly justified by the string field theory. Therefore, a consistent check by both first and second quan-tized approaches of any reliable string results would be of great importance.

The infinitesimal gauge transformation of WSFT is ␦⌽⫽QB⌳⫹g0共⌽*⌳⫺⌳*⌽兲. 共12兲 To compare with our first quantized results in Sec. II, we only need to calculate the first term on the right-hand side 共rhs兲 of Eq. 共12兲. Up to the second massive level, ⌽ and ⌳ can be expressed as ⌽⫽关␾共x兲⫹iA␮共x兲␣⫺1␮ ⫹␣共x兲b⫺1c0⫺B␮␯共x兲␣_⫺1␮ ␣_⫺1␯ ⫹iB␮共x兲␣⫺2␮ ⫹i␤␮共x兲␣␮⫺1b⫺1c0⫹␤0共x兲b⫺2c0 ⫹␤1_共x兲b ⫺1c⫺1⫺iC␮␯␭共x兲␣⫺1␮ ␣⫺1␯ ␣⫺1␭ ⫺C␮␯共x兲␣⫺2␮ ␣⫺1␯ ⫹iC␮共x兲␣⫺3␮ ⫺␥␮␯共x兲␣⫺1␮ ␣⫺1␯ b⫺1c0⫹i␥␮ 0_共x兲␣ ⫺1 ␮ _b ⫺2c0 ⫹i␥␮1共x兲␣⫺1␮ b⫺1c⫺1⫹i␥2␮共x兲␣⫺2␮ b⫺1c0⫹␥0共x兲b⫺3c0 ⫹␥1_共x兲b ⫺2c⫺1⫹␥2共x兲b⫺1c⫺2兴c1兩k典, 共13兲 ⌳⫽关⑀0_共x兲b ⫺1⫺⑀␮␯0 共x兲␣⫺1␮ ␣⫺1␯ b⫺1⫹i⑀␮0共x兲␣⫺1␮ b⫺1 ⫹i⑀_␮1 共x兲␣_⫺2␮ _b ⫺1⫹i⑀␮2共x兲␣⫺1␮ b⫺2⫹⑀1共x兲b⫺2 ⫹⑀2_共x兲b ⫺3⫹⑀3共x兲b⫺1b⫺2c0兴兩⍀典, 共14兲 where ⌽ and ⌳ are restricted to ghost numbers 1 and 0, respectively, and the BRST charge is

Q_B⫽

兺

n⫽⫺⬁ ⬁ L_⫺nmattc_n⫹

兺

m,n⫽⫺⬁ ⬁ _m⫺n 2 :cmcnb⫺m⫺n:⫺c0. 共15兲 The transformations one gets for each mass level are the following: m2⫽0, ␦A_␮⫽⳵_␮⑀0, 共16a兲 ␦ ␣⫽1 2⳵ 2_⑀0_{; 共16b兲} m2⫽2, ␦B_␮␯⫽⫺⳵(␮⑀_␯)0⫺1 2⑀ 1_␩ ␮␯, 共17a兲 ␦B_␮⫽⫺⳵␮⑀1⫹⑀_␮0, 共17b兲 ␦␤␮⫽1 2共⳵ 2_⫺2兲⑀ ␮ 0 , 共17c兲 ␦␤0_⫽1 2共⳵ 2_⫺2兲⑀1_{, 共17d兲} ␦␤1_⫽⫺⳵␮_⑀ ␮ 0_⫺3⑀1_{; 共17e兲} m2⫽4, ␦C_␮␯␭⫽⫺⳵(␮⑀_␯␭)0 ⫺12⑀(␮ 2 _{␩␯␭), 共18a兲} ␦C[␮␯]⫽⫺⳵[␯⑀_␮] 1 _⫺ ⳵[␮⑀␯]2 , 共18b兲 ␦C_(␮␯)⫽⫺⳵(_␯⑀_␮)1 ⫺⳵(␮⑀_␯)2⫹2⑀_␮␯0 ⫺⑀2␩␮␯_{, 共18c兲} ␦C_␮⫽⫺⳵␮⑀2⫹2⑀_␮1⫹⑀_␮2, 共18d兲 ␦␥␮␯⫽1 2共⳵ 2_⫺4兲⑀ ␮␯ 0 _⫺1 2⑀ 3_{␩␮␯, 共18e兲} ␦␥␮0⫽12共⳵2⫺4兲⑀␮ 2_⫹⳵␮⑀3_{, 共18f兲} ␦␥␮1⫽⫺2⳵␯⑀␯␮0 ⫺2⑀␮1⫺3⑀␮2, 共18g兲 ␦␥␮2_⫽1 2共⳵ 2_⫺4兲⑀ ␮ 1_⫺⳵ ␮⑀3, 共18h兲 ␦␥0_⫽1 2共⳵ 2_⫺4兲⑀2_⫺⑀3_{, 共18i兲} ␦␥1_⫽⫺⳵␮_⑀ ␮ 2_⫺4⑀2_⫺2⑀3_{, 共18j兲} ␦␥2_⫽⫺2⳵␮_⑀ ␮ 1_⫺5⑀2_⫹4⑀3_⫹⑀ ␮ 0␮ . 共18k兲

It is interesting to note that Eq. 共16b兲 corresponds to the lifting of the on-mass-shell condition in Eq. 共3b兲. Mean-while, Eqs. 共17c兲 and 共17d兲 correspond to the on-mass-shell condition in Eqs.共5b兲 and 共4b兲, and Eq. 共17e兲 corresponds to the gauge condition in Eq.共4b兲. Similar correspondence ap-plies to level m2⫽4. Equations 共18e兲, 共18f兲, 共18h兲, and 共18i兲 correspond to the on-mass-shell conditions in Eqs.共6b兲, 共7b兲, 共8b兲, and 共9b兲. Equation 共18g兲, 共18j兲, and 共18k兲 correspond to the gauge conditions in Eqs.共6b兲, 共7b兲, and 共8b兲. The trace-less condition in Eq.共6b兲 corresponds to the trace part of Eq. 共18e兲. Also, only zero-norm state transformation parameters appear on the rhs of matter transformation A, B, C, and all ghost transformations correspond, in a one-to-one manner, to the lifting of on-shell conditions 共including on-mass-shell, gauge, and traceless conditions兲 in the OCFQ approach. These important observations simplify the demonstration of decoupling of degenerate positive-norm states at higher mass levels, m2⫽6 and m2⫽8 more specifically, in WSFT, as will be discussed in the rest of this paper.

For m2⫽4, it can be checked that only C_␮␯␭ and C[␮␯] are dynamically independent and they form a gauge multip-let, which is consistent with the result of the first quantized calculation presented in Sec. II.

We now show the decoupling phenomenon for the third massive level m2⫽6, in which ⌽ and ⌳ can be expanded as

⌽4⫽关D␮␯␣␤共x兲␣_⫺1␮ ␣_⫺1␯ ␣_⫺1␣ a_⫺1␤ ⫺iD␮␯␣共x兲␣_⫺1␮ ␣_⫺1␯ ␣_⫺2␣ ⫺D␮␯ 0 _共x兲␣ ⫺2 ␮ _␣ ⫺2 ␯ _⫺D ␮␯ 1 _共x兲␣ ⫺1 ␮ _␣ ⫺3 ␯ _⫹iD ␮共x兲␣⫺4␮ ⫺i␰␮␯␣共x兲␣⫺1␮ ␣⫺1␯ ␣⫺1␣ b⫺1c0⫺␰␮␯ 0 _共x兲␣ ⫺2 ␮ _␣ ⫺1 ␯ _b ⫺1c0⫺␰␮␯ 1 _共x兲␣ ⫺1 ␮ _␣ ⫺1 ␯ _b ⫺2c0⫺␰␮␯ 2 _共x兲␣ ⫺1 ␮ _␣ ⫺1 ␯ _b ⫺1c⫺1 ⫹i␰␮0共x兲␣⫺3␮ b⫺1c0⫹i␰␮ 1_共x兲␣ ⫺2 ␮ _b ⫺2c0⫹i␰␮ 2_共x兲␣ ⫺1 ␮ _b ⫺3c0⫹i␰␮ 3_共x兲␣ ⫺2 ␮ _b ⫺1c⫺1⫹i␰␮4共x兲␣⫺1␮ b⫺2c⫺1 ⫹i␰␮5共x兲␣⫺1␮ b⫺1c⫺2⫹␰0共x兲b⫺4c0⫹␰ 1_共x兲b ⫺3c⫺1⫹␰2共x兲b⫺2c⫺2⫹␰3共x兲b⫺1c⫺3⫹␰4共x兲b⫺2b⫺1c⫺1c0兴c1兩k

典

, 共19兲 ⌳4⫽关⫺i⑀␮␯␣ 0 _共x兲␣ ⫺1 ␮ _␣ ⫺1 ␯ _␣ ⫺1 ␣ _b ⫺1⫺⑀␮␯1 共x兲␣⫺2␮ ␣␯⫺1b⫺1⫺⑀␮␯2 共x兲␣⫺1␮ ␣⫺1␯ b⫺2⫹i⑀␮3共x兲␣⫺3␮ b⫺1⫹i⑀␮4共x兲␣⫺2␮ b⫺2 ⫹i⑀␮5共x兲␣⫺1␮ b⫺3⫹i⑀␮6共x兲␣⫺1␮ b⫺2b⫺1c0⫹⑀4共x兲b⫺4⫹⑀5共x兲b⫺3b⫺1c0⫹⑀6共x兲b⫺2b⫺1c⫺1兴兩⍀典. 共20兲

The transformations for the matter part are the following: ␦D_␮␯␣␤⫽⫺⳵(␤⑀_␮␯␣)0 ⫺12⑀(␮␯ 2 _{␩␣␤), 共21a兲} ␦D_␮␯␣⫽⫺⳵(_␮⑀_{兩␣兩␯)}1 ⫺⳵_␣⑀_␯␮2 ⫹3⑀_␮␯␣0 ⫺12⑀␣ 4_{␩␯␮⫺⑀(} ␮ 5 _␩␯)␣ , 共21b兲 ␦D_[1_␮␯]⫽⫺⳵[_␮⑀_␯]3⫺⳵[_␯⑀_␮]5 ⫹2⑀[1_␯␮], 共21c兲 ␦D₍1_␮␯)⫽⫺⳵(␮⑀_␯)3⫺⳵(␯⑀5_␮)⫹2⑀(1_␯␮)⫹2⑀_␮␯2 ⫺⑀4␩_␮␯, 共21d兲 ␦D_␮␯0 ⫽⫺⳵(_␮⑀_␯)4 ⫹⑀(␮␯)1 ⫺12⑀ 4_{␩␮␯, 共21e兲} ␦D_␮⫽⫺⳵␮⑀4⫹3⑀3_␮⫹2⑀_␮4⫹⑀_␮5 . 共21f兲 It can be checked from Eqs.共21兲 that only D_␮␯␣␤and mixed-symmetric D_␮␯␣cannot be gauged away, which is consistent with the result of the first quantized approach in Sec. II. That is, the spin-two and scalar positive-norm physical propagat-ing modes can be gauged to D_␮␯␣␤ and mixed-symmetric

D_␮␯␣. In fact, D_␮␯␣, D_[1_␮␯], D1_(␮␯), D_␮␯0 , and D_␮ can be gauged away by ⑀_␮␯␭0 , ⑀[1_␮␯], ⑀(␮␯)1 , ⑀_␮␯2 , and one of the vector parameters, say, ⑀␮3_{. The rest, ⑀}

␮ 4_{, ⑀}

␮

5_{, and ⑀}4 _are gauge artifacts of D_␮␯␣␤and mixed-symmetric D_␮␯␣.

The transformations for the ghost part are the following: ␦␰␮␯␣⫽1 2共⳵ 2_⫺6兲⑀ ␮␯␣ 0 _⫺1 2⑀(␮ 6 _␩ ␯␣), 共22a兲 ␦␰[␮␯] 0 _⫽1 2共⳵ 2_⫺6兲⑀[ ␮␯] 1 _⫺ ⳵[␮⑀␯]6 , 共22b兲 ␦␰(␮␯) 0 _⫽1 2共⳵ 2_⫺6兲⑀( ␮␯) 1 _{⫺⳵(␮⑀} ␯) 6_⫹⑀5_{␩␮␯, 共22c兲} ␦␰␮␯1 _⫽1 2共⳵ 2_⫺6兲⑀ ␮␯ 2 _⫹⳵( ␯⑀␮)6 , 共22d兲 ␦␰␮␯2 ⫽⫺3⳵␣⑀␮␯␣0 ⫺2⑀(1␮␯)⫺3⑀␮␯2 ⫺12⑀ 6_␩ ␮␯, 共22e兲 ␦␰␮0⫽12共⳵ 2_⫺6兲⑀ ␮ 3_⫺⳵␮⑀5_⫹⑀ ␮ 6_{, 共22f兲} ␦␰␮1⫽12共⳵ 2_⫺6兲⑀ ␮ 4_⫺⑀ ␮ 6 , 共22g兲 ␦␰␮2⫽12共⳵ 2_⫺6兲⑀ ␮ 5_⫹⳵␮⑀5_⫺⑀ ␮ 6_{, 共22h兲} ␦␰␮3_{⫽⫺⳵␯⑀} ␮␯ 1 _⫺⳵ ␮⑀6⫺3⑀␮3⫺3⑀␮4, 共22i兲 ␦␰␮4⫽2⳵␯⑀␮␯2 ⫹⳵␮⑀6⫺2⑀␮4⫺4⑀␮5⫺2⑀␮6, 共22j兲 ␦␰␮5⫽⫺2⳵␯⑀␯␮1 ⫺3⑀␮3⫺5⑀␮5⫹4⑀␮6⫹3⑀␮␯0␯, 共22k兲 ␦␰0_⫽1 2共⳵ 2_⫺6兲⑀4_⫺2⑀5_{, 共22l兲} ␦␰1_⫽⫺⳵␮_⑀ ␮ 5_⫺5⑀4_⫺2⑀5_⫺⑀6_{, 共22m兲} ␦␰2_⫽⫺2⳵␮_⑀ ␮ 4_⫺6⑀4_⫺3⑀6_⫹⑀ ␮ 2␮_{, 共22n兲} ␦␰3_⫽⫺3⳵␮_⑀ ␮ 3_⫺7⑀4_⫹6⑀5_⫹5⑀6_⫹2⑀ ␮ 1␮_{, 共22o兲} ␦␰4_⫽1 2共⳵ 2_⫺6兲⑀6_⫹⳵␮_⑀ ␮ 6_⫹4⑀5_{. 共22p兲}

There are nine on-mass-shell conditions, which contains a symmetric spin-three, an antisymmetric spin-two, two sym-metric spin-two, three vectors and two scalar fields, and seven gauge conditions, which amount to 16 equations in Eq. 共22兲. This is consistent with counting from zero-norm states listed in the table. Three traceless conditions read from zero-norm states correspond to the three equations involving ␦␰␮␯␯, ␦␰␮0␮, ␦␰␮1␮, which are contained in Eqs. 共22a兲, 共22c兲, and 共22d兲.

It is important to note that the transformation for the mat-ter parts, Eqs. 共18a兲–共18d兲 and Eqs. 共21a兲–共21f兲, are the same as the calculation关5兴 based on the chordal gauge trans-formation of free covariant string field theory constructed by Banks and Peskin 关10兴. The chordal gauge transformation can be written in the following form:

␦⌽关X共␴兲兴⫽

兺

n⬎0

L_⫺n⌽_n关X共␴兲兴, 共23兲

where⌽关X(␴)兴 is the string field and ⌽n关X(␴)兴 are gauge

parameters, which are functions of X关␴兴 only and free of ghost fields. This is because the pure ghost part of QBin Eq.

共15兲 does not contribute to the transformation of matter back-ground fields. It is interesting to note that the rhs of Eq.共23兲 is in the form of off-shell spurious states关11兴 in the OCFQ approach. They become zero-norm states on imposing the physical and on-shell state condition.

Finally, it can be shown that the number of scalar zero-norm states at nth massive level (n⭓3) is at least the sum of those at (n⫺2)th and (n⫺1)th massive levels. So positive-norm scalar modes at nth level, if they exist, will be decou-pled according to our decoupling conjecture. The decoupling of these scalars has important implication on Sen’s conjec-tures on the decay of open string tachyon. Since all scalars on D-brane including tachyon get a nonzero VEV in the false vacuum, they will decay together with tachyon and disappear eventually to the true closed string vacuum. As the scalar states together with the higher tensor states form a large gauge multiplet at each mass level, and its scattering ampli-tudes are fixed by the tensor fields, these tensor fields of open string共D25-brane兲 will accompany the decay process. This means that the whole D-brane could disappear to the true closed string vacuum! The mechanism could provide a hint to solve the so-called U(1) problem 关19兴 in Sen’s con-jectures. A further study is in progress.

IV. THE FOURTH MASSIVE LEVEL

We will use both the first and the second quantized ap-proaches to test the decoupling conjecture for the fourth mas-sive level m2⫽8.

A. The first quantized calculation

The positive-norm physical propagating fields can be found in Ref. 关20兴. Their Young tabulations are the follow-ing:

共24兲

The Young tabulations of zero-norm states can then be shown to be 共25兲 ␣_⫺1␮ _␣ ⫺2 ␯ _␣ ⫺2

␭ _{and the other corresponds to ␣} ⫺1 ␮ _␣ ⫺1 ␯ _␣ ⫺3 ␭ _or

vice versa. So, one expects that the last three states in Eq.

共24兲 can be gauged to the higher rank fields. The most

gen-eral worldsheet coupling consistent with vertex operator con-sideration is T共␶兲⫽⫺12␩␮␯⳵␶X␮⳵␶X␯ ⫹E␮␯␭␣␤⳵␶X␮⳵␶X␯⳵␶X␭⳵␶X␣⳵␶X␤ ⫹E␮␯␭␣⳵␶X␮⳵␶X␯⳵␶X␭⳵␶2X␣ ⫹E␮␯␭0 ⳵␶X␮⳵␶X␯⳵␶3X␭⫹E␮␯␭1 ⳵␶X␮⳵␶2X␯⳵␶2X␭ ⫹E_␮␯0 _⳵ ␶X␮⳵␶4X␯⫹E␮␯1 ⳵␶2X␮⳵␶3X␯⫹E␮⳵␶5X␮. 共26兲

After a lengthy calculation, the condition to cancel all world-sheet q number anomalies are as follows:

5⳵␮E_{␮␯␭␣␤}⫺2E(␯␭␣␤)⫽0, 共27a兲 ⳵␮_E ␮␯ 0 _⫺20E ␯⫽0, 共27b兲 ⳵␮_E ␮␯␭␣⫺12E␯␭␣0 ⫺8E␯␭␣1 ⫽0, 共27c兲 ⳵␮_E ␮␯␭ 0 _⫺6E ␯␭ 0 _⫺E ␯␭ 1 _{⫽0, 共27d兲} ⳵␮_E ␮␯␭ 1 _⫺6E (␯␭) 1 _{⫽0, 共27e兲} 20E␮_␮␯␭␣⫹⳵␮E_␯␭␣␮⫺12E(␯␭␣) 0 _{⫽0, 共28a兲} E0␮_␮␯⫹4⳵␮E_␮␯1 ⫺120E_␯⫽0, 共28b兲 E␮_␮␯␭⫹8⳵␮E1_␯␭␮⫺48E_␯␭0 ⫺12E_␭␯1 ⫽0, 共28c兲 E␮_␯␭␮⫹⳵␮E_␯␭␮0 ⫺4E₍0_␯␭)⫽0, 共29a兲 E1␮_␮␯⫹12⳵␮E_␯␮1 ⫺240E_␯⫽0, 共29b兲 3E0␮_␯␮⫹E_␯␮1␮⫹6⳵␮E_␯␮0 ⫺30E_␯⫽0, 共30兲 2E0␮ ␮⫹E␮1␮⫹10⳵␮E␮⫽0, 共31兲 共⳵2_{⫺8兲␾⫽0. 共32兲}

Here,␾ again represents all background fields introduced in Eq. 共26兲. Equations 共27a兲–共27e兲 are extracted from 1/(␶

⫺␶

⬘

)3 anomalous terms in the operator product calculation; similarly, Eqs. 共28a兲–共28c兲, 共29a兲–共29b兲, 共30兲, and 共31兲 are extracted from 1/(␶⫺␶

⬘

)4, 1/(␶⫺␶

⬘

)5, 1/(␶⫺␶

⬘

)6, and 1/(␶⫺␶

⬘

)7anomalous terms, respectively. It can be carefully checked, as one did for the third massive level, that only

E_{␮␯␭␣␤} and mixed-symmetric E_␮␯␭␣ and E_␮␯␭1 共or E_␮␯␭0 ) corresponding to the first three Young representations in Eq.

共24兲 are dynamically independent as the conjecture has

claimed. The last three states in Eq.共24兲 again can be gauged to the first three states due to the existence of zero-norm states with the same Young representations in Eq. 共25兲.

B. WSFT calculation

⌽ and ⌳ can be expanded at this massive level as

⌽5⫽关iE␮␯␭␣␤共x兲␣_⫺1␮ ␣_⫺1␯ ␣_⫺1␣ a_⫺1␭ a_⫺1␤ ⫹E␮␯␣␤共x兲␣_⫺1␮ ␣␯_⫺1␣_⫺1␣ a_⫺2␤ ⫺iE␮␯␣ 0 _共x兲␣ ⫺1 ␮ _␣ ⫺1 ␯ _␣ ⫺31 ␣ _⫺iE ␮␯␣ 1 _共x兲␣ ⫺1 ␮ _␣ ⫺2 ␯ _␣ ⫺2 ␣ ⫺E␮␯0 共x兲␣⫺1␮ ␣⫺4␯ ⫺E1␮␯共x兲␣⫺2␮ ␣⫺3␯ ⫹iE␮共x兲␣⫺5␮ ⫹␨␮␯␣␤共x兲␣⫺1␮ ␣⫺1␯ ␣⫺1␣ ␣⫺1␤ b⫺1c0⫺i␨␮␯␣

0 _共x兲␣ ⫺2 ␮ _␣ ⫺1 ␯ _␣ ⫺1 ␣ _b ⫺1c0 ⫺i␨␮␯␣1 共x兲␣⫺1␮ ␣␯⫺1␣⫺1␣ b⫺2c0⫺i␨␮␯␣ 2 _共x兲␣ ⫺1 ␮ _␣ ⫺1 ␯ _␣ ⫺1 ␣ _b ⫺1c⫺1⫺␨␮␯0 共x兲␣␮⫺3␣⫺1␯ b⫺1c0⫺␨␮␯ 1 _共x兲␣ ⫺2 ␮ _␣ ⫺2 ␯ _b ⫺1c0 ⫺␨␮␯2 共x兲␣⫺2␮ ␣⫺1␯ b⫺2c0⫺␨␮␯ 3 _共x兲␣ ⫺1 ␮ _␣ ⫺1 ␯ _b ⫺3c0⫺␨␮␯ 4 _共x兲␣ ⫺2 ␮ _␣ ⫺1 ␯ _b ⫺1c⫺1⫺␨␮␯5 共x兲␣⫺1␮ ␣⫺1␯ b⫺2c⫺1 ⫺␨␮␯6 共x兲␣⫺1␮ ␣⫺1␯ b⫺1c⫺2⫹i␨␮0共x兲␣⫺4␮ b⫺1c0⫹i␨␮ 1_共x兲␣ ⫺3 ␮ _b ⫺2c0⫹i␨␮ 2_共x兲␣ ⫺2 ␮ _b ⫺3c0⫹i␨␮ 3_共x兲␣ ⫺1 ␮ _b ⫺4c0 ⫹i␨␮4共x兲␣⫺3␮ b⫺1c⫺1⫹i␨5␮共x兲␣⫺2␮ b⫺2c⫺1⫹i␨␮6共x兲␣⫺1␮ b⫺3c⫺1⫹i␨␮7共x兲␣⫺2␮ b⫺1c⫺2⫹i␨␮8共x兲␣⫺1␮ b⫺2c⫺2 ⫹i␨␮9共x兲␣⫺1␮ b⫺1c⫺3⫹i␨␮10共x兲␣⫺1␮ b⫺2b⫺1c⫺1c0⫹␨0共x兲b_⫺5c0⫹␨1共x兲b_⫺4c_⫺1⫹␨2共x兲b_⫺3c_⫺2⫹␨3共x兲b_⫺2c_⫺3 ⫹␨4_共x兲b

⫺1c⫺4⫹␨5共x兲b⫺3b⫺1c⫺1c0⫹␨6共x兲b⫺2b⫺1c⫺2c0兴c1兩k

典

, 共33兲 ⌳5⫽关⑀_␮␯␣␤0 共x兲␣_⫺1␮ ␣_⫺1␯ ␣_⫺1␣ ␣_⫺1␤ b_⫺1⫺i⑀_␮␯␣1 共x兲␣_⫺2␮ ␣_⫺1␯ ␣_⫺1␣ b_⫺1⫺i⑀_␮␯␣2 共x兲␣_⫺1␮ ␣_⫺1␯ ␣␣_⫺1b_⫺2⫺⑀_␮␯3 共x兲␣_⫺3␮ ␣_⫺1␯ b_⫺1

⫺⑀␮␯4 共x兲␣⫺2␮ ␣⫺2␯ b⫺1⫺⑀5␮␯共x兲␣⫺2␮ ␣⫺1␯ b⫺2⫺⑀␮␯6 共x兲␣␮⫺1␣⫺1␯ b⫺3⫺⑀␮␯7 共x兲␣⫺1␮ ␣␯⫺1b⫺2b⫺1c0⫹i⑀_␮7共x兲␣_⫺4␮ b_⫺1 ⫹i⑀␮8共x兲␣⫺3␮ b⫺2⫹i⑀␮9共x兲␣⫺2␮ b⫺3⫹i⑀␮10共x兲␣⫺1␮ b⫺4⫹i⑀␮11共x兲␣⫺2␮ b⫺2b⫺1c0⫹i⑀_␮12共x兲␣_⫺1␮ b⫺3b⫺1c0

⫹i⑀_␮13

共x兲␣_⫺1␮ _b

⫺2b⫺1c⫺1⫹⑀7共x兲b⫺5⫹⑀8共x兲b⫺4b⫺1c0⫹⑀9共x兲b⫺3b⫺2c0⫹⑀10共x兲b⫺3b⫺1c⫺1 ⫹⑀11_共x兲b

⫺2b⫺1c⫺2兴兩⍀典. 共34兲

The transformations for the matter part are ␦E_{␮␯␭␣␤}⫽⫺⳵(_␤⑀_{␮␯␭␣)}0 ⫹12⑀(␭␣␤ 2 _␩␮␯) , 共35a兲 ␦E_␮␯␣␤⫽⫺⳵(_␮⑀_{兩␤兩␣␯)}1 ⫺⳵␤⑀_␣␮␯2 ⫹4⑀_␮␯␣␤0 ⫺12⑀␤(␯ 5 _␩␣␮) ⫺⑀(6_{␣␮␩␯)␤} , 共35b兲 ␦E_␮␯␣0 ⫽⫺⳵(␮⑀兩␣兩␯)3 ⫺⳵␣⑀_␯␮6 ⫹2⑀_␣␯␮1 ⫹3⑀_␣␯␮2 ⫺12⑀␣ 7_␩ ␯␮ ⫺⑀(9_{␮␩␯)␣} , 共35c兲 ␦E_␮␯␣1 ⫽⫺⳵␮⑀_␯␣4 ⫺⳵(_␣⑀_␯)␮5 ⫹2⑀(1_␣␯)␮⫺⑀(8_␣␩␯)␮ ⫺1 2⑀␮ 9_␩ ␯␣, 共35d兲 ␦E_[0_␮␯]⫽⫺⳵[␮⑀_␯]7⫺⳵[␯⑀_␮]9 ⫹3⑀[3_␯␮]⫹2⑀[5_␯␮], 共35e兲 ␦E_[1_␮␯]⫽⫺⳵[␮⑀_␯]7⫺⳵[_␯⑀_␮]8 ⫹⑀[3_␯␮]⫹⑀[5_␮␯], 共35f兲 ␦E(␮␯) 0 _⫽⫺⳵( ␮⑀␯)7⫺⳵(␯⑀␮)9 ⫹3⑀(3␯␮)⫹2⑀(5␯␮)⫹2⑀␯␮6 ⫺⑀7_{␩␮␯, 共35g兲} ␦E₍1_␮␯)⫽⫺⳵(␮⑀_␯)7⫺⳵(_␯⑀_␮)8 ⫹⑀(3_␯␮)⫹2⑀_␯␮4 ⫹⑀(5_␮␯) ⫺⑀7_{␩␮␯, 共35h兲} ␦E_␮⫽⫺⳵␮⑀7⫹7⑀7_␮⫹2⑀_␮8⫹⑀_␮9. 共35i兲 Again these are the same as the calculation by Eq. 共23兲. All background fields except E_{␮␯␭␣␤} and mixed-symmetric

E_␮␯␭␣ and E_␮␯␭1 共or E_␮␯␭0 ) can be either gauged away or gauged to E_{␮␯␭␣␤}, E_␮␯␭␣, and E_␮␯␭1 共or E_␮␯␭0 ) by zero-norm states. This is consistent with the result of the first quantized approach presented in Sec. IV A. The transforma-tion for the ghost part is very lengthy and is given in the Appendix. There are 18 on-mass-shell conditions, which contain a spin-four, a mixed-symmetric spin-three, two sym-metric spin-three, two antisymsym-metric spin-two, four symmet-ric spin-two, five vector and three scalar fields, and 15 gauge conditions. It is again consistent with counting the number of zero-norm states listed in Eq.共25兲.

V. CONCLUSION

We have explicitly shown that the degenerate positive-norm states at the third and fourth massive levels of bosonic open string theory can be gauged to the higher rank fields at the same mass level. This means that the scattering ampli-tudes of these degenerate positive-norm states can be ex-pressed in terms of those of higher spin states at the same mass level through massive Ward identities. This is demon-strated by using both the OCFQ string and WSFT. We have compared the on-shell conditions of zero-norm states in the OCFQ stringy gauge transformation to the background ghost fields in off-shell gauge transformation of WSFT. This im-portant observation makes the lengthy calculations in both the first and the second quantized approaches controllable and more importantly provides a double consistency check of our results. The interesting stringy behaviors discussed in this paper and those in Refs. 关1,2兴 seem to imply that there

must exist enormous exotic high-energy properties of string theory, which remained to be uncovered. One interesting ap-plication of the decoupling of higher scalar modes is the decay of tensor fields on D-brane into the true closed string vacuum in Sen’s conjectures.

It is straightforward to generalize our calculation to closed string theory for the first quantized approach presented in Secs. II and IV A. Another way to generalize to the closed string case is to make use of the simple relation between closed and open string amplitudes in Ref. 关21兴. A reliable second quantized closed string field theory may help uncover more high-energy stringy properties.

ACKNOWLEDGMENTS

The authors would like to thank Pei-ming Ho for stimu-lating discussions, which help clarify some of the points in the paper. The authors also thank Miao Li for helpful con-versations. This work was supported in part by the National Science Council, the Center for Theoretical Physics at Na-tional Taiwan University, the NaNa-tional Center for Theoretical Sciences, and the CosPA project of the Ministry of Educa-tion, Taiwan, Republic of China.

APPENDIX

Gauge transformation for background ghost fields of the fourth massive level are as follows:

␦␨␮␯␣␤⫽12共⳵ 2_⫺8兲⑀ ␮␯␣␤ 0 _⫺1 2⑀(␮␯ 7 _␩␣␤) , 共A1兲 ␦␨␮␯␣0 ⫽12共⳵ 2_⫺8兲⑀ ␮␯␣ 1 _⫺ ⳵␮⑀_␯␣7 _⫺1 2⑀␮ 11_{␩␯␣⫺⑀(} ␯ 12_␩ ␣)␮, 共A2兲 ␦␨␮␯␣1 ⫽12共⳵ 2_⫺8兲⑀ ␮␯␣ 2 _⫹ ⳵(␮⑀␯␣)7 , 共A3兲 ␦␨␮␯␣2 ⫽⫺4⳵␤⑀␮␯␣␤0 ⫺2⑀(␮␯␣)1 ⫺3⑀␮␯␣2 ⫺12⑀(␮ 13_␩ ␯␣), 共A4兲 ␦␨[␮␯] 0 _⫽1 2共⳵ 2_⫺8兲⑀[ ␮␯] 3 _⫺⳵[ ␮⑀␯]12, 共A5兲 ␦␨[␮␯] 2 _⫽1 2共⳵ 2_⫺8兲⑀[ ␮␯] 5 _⫹⳵[ ␯⑀␮]11, 共A6兲 ␦␨[␮␯] 4 _⫽⫺2⳵␣_⑀[ ␮␯]␣ 1 _{⫺⳵[␮⑀} ␯] 13_⫺3⑀[ ␮␯] 3 _⫺3⑀[ ␮␯] 5 _, 共A7兲 ␦␨(␮␯) 0 _⫽1 2共⳵ 2_⫺8兲⑀( ␮␯) 3 _{⫺⳵(␮⑀} ␯) 12_⫹2⑀ ␮␯ 7 _⫺⑀8_␩ ␮␯, 共A8兲 ␦␨␮␯1 ⫽12共⳵ 2_⫺8兲⑀ ␮␯ 4 _{⫺⳵(␮⑀} ␯)11⫺12⑀ 8_␩ ␣␤, 共A9兲 ␦␨(␮␯) 2 _⫽1 2共⳵ 2_⫺8兲⑀( ␮␯) 5 _⫹⳵( ␯⑀␮)11⫺2⑀␮␯7 ⫺⑀9␩␮␯, 共A10兲 ␦␨␮␯3 ⫽12共⳵ 2_⫺8兲⑀ ␮␯ 6 _⫹ ⳵(␮⑀␯)12⫺⑀␮␯7 ⫹12⑀ 9_␩␮␯, 共A11兲 ␦␨(␮␯) 4 _⫽⫺2 ⳵␣_⑀( ␮␯)␣ 1 _⫺ ⳵(␮⑀␯)13⫺3⑀(3␮␯)⫺4⑀␮␯4 ⫺3⑀(5␮␯) ⫺⑀10_{␩␮␯, 共A12兲} ␦␨␮␯5 _{⫽⫺3⳵␣⑀} ␮␯␣ 2 _⫹⳵( ␮⑀␯)13⫺2⑀(5␮␯)⫺4⑀␮␯6 ⫺2⑀␮␯7 , 共A13兲 ␦␨␮␯6 _{⫽⫺2⳵␣⑀} ␣␮␯ 1 _⫹6⑀ ␮␯␣␤ 0 _{␩␣␤⫺3⑀(␮␯)}3 _⫺5⑀ ␮␯ 6 _⫹4⑀ ␮␯ 7 ⫺1 2⑀ 11_{␩␮␯, 共A14兲} ␦␨␮0⫽12共⳵ 2_⫺8兲⑀ ␮ 7_⫺⳵␮⑀8_⫹2⑀ ␮ 11_⫹⑀ ␮ 12_{, 共A15兲} ␦␨␮1_⫽1 2共⳵ 2_⫺8兲⑀ ␮ 8_⫺⳵ ␮⑀9⫺2⑀␮11, 共A16兲 ␦␨␮2_⫽1 2共⳵ 2_⫺8兲⑀ ␮ 9_⫹⳵ ␮⑀9⫺⑀␮12, 共A17兲 ␦␨␮3_⫽1 2共⳵ 2_⫺8兲⑀ ␮ 10_⫹⳵ ␮⑀8⫺2⑀␮12, 共A18兲 ␦␨␮4_{⫽⫺⳵␯⑀} ␮␯ 3 _⫺⳵ ␮⑀10⫺4⑀␮7⫺3⑀␮8⫹⑀␮13, 共A19兲 ␦␨␮5⫽⫺⳵␯⑀␮␯5 ⫺3⑀␮8⫺4⑀␮9⫺2⑀␮11⫺⑀␮13, 共A20兲 ␦␨␮6⫽⫺2⳵␯⑀␮␯6 ⫹⳵␮⑀10⫺2⑀␮9⫺5⑀␮10⫺2⑀␮12⫺⑀␮13, 共A21兲 ␦␨␮7⫽⫺4⳵␯⑀␮␯4 ⫺⳵␮⑀11⫺4⑀7␮⫺5⑀␮9⫹4⑀␮11⫹⑀␮␯␣1 ␩␯␣, 共A22兲 ␦␨␮8⫽⫺2⳵␯⑀␯␮5 ⫹⳵␮⑀11⫺3⑀8␮⫺6⑀␮10⫺3⑀␮13⫹3⑀␮␯␣2 ␩␯␣, 共A23兲 ␦␨␮9⫽⫺3⳵␯⑀␮␯3 ⫺4⑀␮7⫺7⑀10␮⫹6⑀␮12⫹5⑀␮13⫹4⑀␯␣␮1 ␩␯␣, 共A24兲 ␦␨␮10⫽12共⳵ 2_⫺8兲⑀ ␮ 13_⫹2 ⳵␯_⑀ ␮␯ 7 _⫹2⑀ ␮ 11_⫹4⑀ ␮ 12 , 共A25兲 ␦␨0_⫽1 2共⳵ 2_⫺8兲⑀7_⫺3⑀8_⫺⑀9_{, 共A26兲} ␦␨1_⫽⫺⳵␮_⑀ ␮ 10_⫺6⑀7_⫺2⑀8_⫺2⑀10_{, 共A27兲} ␦␨2_⫽⫺⳵␮_⑀ ␮ 9_⫺7⑀7_⫺4⑀9_⫺3⑀10_⫺⑀11_⫹⑀ ␮␯ 6 _␩␮␯_, 共A28兲 ␦␨3_⫽⫺3⳵␮_⑀ ␮ 8_⫺8 ⑀7_⫹6⑀9_⫺4⑀11_⫹2⑀ ␮␯ 5 _␩␮␯ , 共A29兲 ␦␨4_⫽⫺4⳵␮_⑀ ␮ 7_⫺9⑀7_⫹8⑀8_⫹7⑀10_⫹6⑀11_⫹3⑀ ␮␯ 3 _␩␮␯ ⫹4⑀_␮␯4 _␩␮␯ , 共A30兲 ␦␨5_⫽1 2共⳵ 2_⫺8兲⑀10_⫹⳵␮_⑀ ␮ 12_⫹5⑀8_⫹3⑀9_{, 共A31兲} ␦␨6_⫽1 2共⳵ 2_⫺8兲⑀11_⫹2⳵␮_⑀ ␮ 11_⫹6⑀8_⫺5⑀9_⫺⑀ ␮␯ 7 _␩␮␯ . 共A32兲 There are 18 on-mass-shell conditions and 15 gauge condi-tions in Eqs.共A1兲–共A32兲, which are consistent with counting from number of zero-norm states listed in Eq.共25兲. Note that there are two irreducible components in Eq.共A2兲.

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