QCD state

Indications of the possible observation of the lowest-lying 1�� QCD state

Kwei-Chou YangDepartment of Physics, Chung-Yuan Christian University, Chung-Li, Taiwan 320, Republic of China

(Received 17 March 2007; published 1 November 2007)

We discuss properties of 1�� exotic mesons within the framework of the QCD field-theoretic approach.We estimate the mass of the lowest-lying 1�� exotic meson using renormalization-improved QCD sumrules, and find that the mass lies around 1:26� 0:15 GeV, in good agreement with the �1�1400� data. Thisstate should be expected in QCD. We find that the mass for the lowest-lying strange 1�� meson is 1:31�0:19 GeV. Our result hints that the K��1410� may be the lowest-lying 1�� nonet state.

DOI: 10.1103/PhysRevD.76.094001 PACS numbers: 12.39.Mk, 11.55.Hx, 12.38.Aw, 12.38.Lg

I. INTRODUCTION

Quantum chromodynamics (QCD) is the presently ac-cepted theory of strong interactions among quarks andgluons. Perturbative QCD has been developed in greatdetail and tested successfully. However, the physics ofthe nonperturbative QCD seems to be a much more diffi-cult task. The rapid development of the lattice QCD theorymay offer some answers to the nonperturbative dynamics.

In addition to normal mesons ( �qq) and baryons (qqq),which can be built up from the naive quark model, QCDallows the existence of exotic states which can be glue-balls, hybrid mesons (the bound states of �qqg), tetraquarkmesons (the bound states of �qq �qq), and other multipartonstates. These exotic states, which go beyond the descriptionof the naive quark model, can offer direct evidence con-cerning confining properties of QCD.

In the past years possible evidences have accumulatedfor the existence of 1�� exotic states [1]. Now we have two1�� exotic states below 2 GeV. A negatively chargedexotic state, �1�1400�, with JPC � 1�� was observed in��p! ��p [2,3] and in �pn! �0�� [4]. The corre-sponding neutral state was reported by the Crystal Barrel[5] and E852 [6] collaborations in the reactions of �pp!�0�0� and ��p! ��0n, respectively, where the decaychannel �� is an isovector and hence cannot be confusedwith a glueball. Unlike the charged �� channel, thecharge conjugate is a good quantum number in the neutral��0 system, where the � was detected in its 2� decaymode [6] and very recently in its ��0 decay mode inthe E852 experiment [7]. The advantage of detecting the�! ��0 mode over the all-neutral final state is thatthe decaying vertex is determined by charged tracks. Themass of the neutral exotic 1�� state observed in the veryrecent E852 experiment is 1257� 20� 25 MeV [7],which is lower than the mass 1360� 25 MeV in theCrystal Barrel measurement. The world average is 1376�19 MeV [1].

Another observed 1�� exotic state, �1�1600�, has alsoattracted much attention. In contrast to �1�1400�, the �0�coupling of this state is unexpectedly stronger than the ��coupling, although the phase space favors the latter.Because of the above property, it was argued that

�1�1400� may be favored for a four-quark state, while�1�1600� may be a hybrid meson [8].

Theoretically, the compositions of observed 1�� exoticstates remain unclear. The flux-tube model [8] and latticecalculations [9] predict the lowest-lying 1�� hybrid mesonto have a mass of about 1.9 GeV. Moreover, the formerpredicts that the 1�� hybrid meson is dominated by b1�and f1� decays in contrast with the�1�1600� experimentalresults where the three final states b1�, �0�, and �� are ofcomparable strength, 1:1:0� 0:3:1:5� 0:5 [10].

To evaluate the mass of the lowest-lying 1�� hybridmeson in the QCD sum rule approach, in the literature

J��x� � i �d�x��gsG��x�u�x�

was used as the relevant interpolating current in the calcu-lation of the two-point correlation function. The earliestQCD sum rule calculations were done in Refs. [11,12],where the radiative corrections and anomalous dimensionsof J� and some operators in operator-product-expansion(OPE) series were not known. To obtain a more preciseestimate, the radiative corrections, which are sizable, werethen computed in Refs. [13,14]. Unfortunately, the masssum rule result is highly sensitive to the parameter s0 whichmodels the threshold of the excited states and does notshow s0 stability. Thus, Jin and Korner conservativelyestimate that the mass of the lowest-lying 1�� hybridmeson is larger than 1.55 GeV [14]. Nevertheless,Chetyrkin and Narison argue that s0 may lie between3:5 GeV2 and 4:5 GeV2, so that the mass of the lowest-lying 1�� hybrid meson is about 1:6� 1:7 GeV [13].

To clarify the inconsistencies in the literature, in thepresent work we study the mass of the lowest-lying 1��

hybrid meson using the QCD sum rule technique. Insteadof J�, we will use

J�x� � n�n� �d�x��gsG��x�u�x�:

Here and in what follows, n� is a lightlike vector, whichsatisfies n2 � 0. So far no one has adopted the abovecurrent to study the mass of the so-called hybrid meson.The reason that we use J is because the residue for Jcoupling to the 1�� hybrid meson determines the normal-ization of the twist-3 light-cone distribution amplitude

PHYSICAL REVIEW D 76, 094001 (2007)

1550-7998=2007=76(9)=094001(7) 094001-1 © 2007 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.76.094001

(LCDA) of the lowest-lying 1�� meson, while for J� theresidue corresponds to the normalization of twist-4 LCDA.For a relevant reaction, the amplitude related to J� isrelatively suppressed by m=Q, as compared with that dueto J, where Q corresponds to the scale of the reaction andm is the mass of the 1�� meson. Thus J may be moresuitable to use than J� to study the lowest-lying 1��

meson.Note that the renormalization (RG)-improvement is al-

ways necessary in the method of QCD sum rules becausethe hadronic mass is determined by the maximum stabilityof the sum rule within the Borel window, where the Borelmass is actually the renormalization scale. Unlike thevector current �q��q which satisfies the current conserva-tion and is scale independent, J, as well as J�, is a scale-dependent operator, similar to the baryonic cases [15,16],and its anomalous dimension may result in significantimprovement in the QCD sum rule result.

The rest of this paper is organized as follows. In Sec. II,we make a brief description of LCDAs for the 1�� exoticmeson in the language of the QCD field theory. TheLCDAs play an essential role in the QCD description ofhard exclusive processes. Using the QCD sum rule tech-nique and adopting the current J, in Sec. III we are devotedto the calculation of the mass sum rule of the lowest-lying1�� meson. To clarify the discrepancy between our resultand those given in the literature, we revisit the previousQCD sum rule studies in Sec. IV, where the current J� isadopted. Finally, Sec. V contains the conclusions anddiscussions.

II. THE FRAMEWORK OF QCD

Unlike the model-building approach, a 1�� exotic me-son in the language of QCD field theory is described interms of a set of Fock states for which each state has thesame quantum number as the exotic meson:

j1��i � q �qj �qqi � q �qgjq �qgi � q �qq �qjq �qq �qi � . . . ;

(1)

where i are distribution amplitudes and the dots denotethe higher Fock states. In contrast to the usual intuition ofpeople, actually a nonlocal q �q does not vanish and isantisymmetric under interchange of momentum fractionsof �q and q in the SU(3) limit. For instance, projecting alowest-lying 1�� meson (�1) along the light cone (z2 �0), the leading-twist LCDAs k;? are defined as

h0j �q1�z�z6 q2�0�j�1�P; �i � m2�1�� zk�;

h0j �q1�z��?�z�q2�0�j�1�P; �i � im�1

��? p z?�;(2)

where the notations k;?� �R

10 due

�i �upzk;?�u� andp� � P� � z�m2

�1=�2pz� are introduced, with �u�u� being

the momentum fraction carried by �q1�q2�, and the nonlocalquark-antiquark pair, connected by the Wilson line (notshown), is at lightlike separation. Considering the G-parityin Eq. (2), it can be known that k;? are antisymmetricunder interchange u$ �u in the SU(3) limit; i.e., the am-plitudes vanish in the z! 0 limit (but not in the z2 ! 0limit). This property was first used in the study of the deepexclusive electroproduction involving a lowest-lying 1��

exotic meson [17]. In analogy to the leading LCDAs, alltwist-3 two-parton LCDAs for a 1�� exotic meson are alsoantisymmetric under interchange u$ �u in the SU(3) limitdue to the G-parity. q �qg can be nonvanishing under the interchange of

momentum fractions of quarks. Similar to the case ofvector mesons [18], using nonlocal, three-parton, gauge-invariant operators to project amplitudes of the jq �qgi, wehave three twist-3 three-parton LCDAs for a 1�� exoticmeson, built up by a quark, an antiquark, and a gluon,where two of the LCDAs are symmetric under the inter-change of momentum fractions of the quark and antiquarkin the SU(3) limit, while one is antisymmetric.

III. EVALUATION OF THE MASS OF THELOWEST-LYING 1�� MESON

Adopting the local gauge-invariant current1

J�x� � n�n� �d�x��gsG��x�u�x�; (3)

we shall employ the QCD sum rules [19] to evaluate themass of the lowest-lying 1�� exotic meson. J�x� isG-parity odd, the same as the 1�� isovector state. Theresidue of J coupled to the 1�� state is defined as

h0jJ�0�j1��p; �i � f?3;1��m1��"�� n��p n�: (4)

The residue constant determines the normalization of oneof the twist-3 three-parton LCDAs, and is also the coeffi-cient with conformal spin 7=2 in the conformal partialwave expansion for the LCDA [20].

The method of the QCD sum rule approaches the boundstate problem in QCD from the perturbative region, wherenonperturbative quantities, such as some condensates, maycontribute significant corrections in the OPE series.Through the study of the relevant correlation functionand the idea of the quark-hadron duality, the correspondinghadronic properties, like masses, decay constants, formfactors, etc., can thus be obtained.

We consider the two-point correlation function

iZd4xeiqxh0jTJ�x�Jy�0�j0i � ��q2��q n�4: (5)

1The other possible choice of the twist-3 current is to considern�n� �d�x�gs��G��x�u�x�. One can also consider the four-quarkoperator relevant to the jq �qq �qi Fock state; however, the resultingsum rule will be clouded by the factorization of condensates.

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It should be noted that J�x� can couple not only to 1��

states but also to 0�� states as

h0jJ�0�j0��p; �i � �2f3;S�p n�2; (6)

where the lowest-lying state in the 0�� channel is a0�980�.Therefore, to extract the lowest-lying meson correspond-ing to the 1�� channel, at the hadron level of Eq. (5) weshall consider two lowest-lying states. Our final result willindicate that one of the two lowest-lying states is a0�980�and the other one is the lowest-lying 1�� meson; i.e., themass of the latter is lower than that of a0�1450�, the firstexcited state in the 0�� channel. We approximate thecorrelation function as

4�f3;a0�2

m2a0� q2 �

�f?3;�1�2

m2�1� q2 �

1

�

Z s0

0ds

Im�OPE

s� q2 ; (7)

where �OPE is the OPE result at the quark-gluon level, ands0 is the threshold of higher resonances. We apply theBorel transformation to both sides of Eq. (7) to improvethe convergence of the OPE series and to suppress contri-butions from higher resonances. The sum rule for the firsttwo lowest-lying states can be written as

4e�m2a0=M2

f23;a0� e�m

2�1=M2

�f?3;�1�2

�1

�

Z s0

0dse�s=M

2Im�OPE�s�; (8)

whereM is the so-called Borel mass. We have checked thatour �OPE result is consistent with that given in Ref. [21],where the authors combine �OPE with another correlationfunction result for calculating the twist-3 parameter ofpseudoscalar mesons. Note that the authors of Ref. [21]did not perform the sum rule analysis as we do here. Toimprove further the M2 range of the derived QCD sumrules, we consider the M2 dependence of the various termsusing the RG equation. Thus we need to multiply each OPEterm by a coefficient L��2�J��n�=2b, where L ��s�M�=�s��, �J � 2�7CF=3� Nc� is the anomalous di-mension of J, �n is the anomalous dimension of thecorresponding operator, and b � �11Nc � 2nf�=3 withNc and nf being numbers of colors and flavors, respec-tively [22]. The anomalous dimensions of operators ap-pearing in the OPE series can be found in Ref. [16]. Thefull RG-improved sum rule thus reads

4e�m2a0=M2

f3;a0��2 � e�m

2�1=M2

f?3;�1��2 �

��s��

360�3

Z s0

0se�s=M

2ds�

89

5184

�s��

�2 h�sG2i� �

�s��18�

�muh �uui�

�mdh �ddi�� s��108�

1

M2 �muh �ugs�Gui� �mdh �dgs�Gdi��L14=3b

�71

729

�s��2

M2 �h �uui2� � h �ddi2��L

�b�8�=b

�32

81

�s��2

M2 h �uui�h �ddi�L�b�8�=b

�L�b��J�=b: (9)

To subtract the contribution arising from the lowest-lying scalar meson, we further evaluate the following nondiagonalcorrelation function,

iZd4xeiqxh0jTJ�x� �u�0�d�0�j0i � ��q2��q n�2; (10)

where the scalar current can couple to the lowest-lying scalar meson a0�980�:

h0j �u�0�d�0�ja0�980�i � ma0fa0: (11)

A similar nondiagonal correlation function was carried out in Ref. [21], where there are additional �5’s in two currents.Similarly, we thus obtain

2e�m2a0=M2

f3;a0��fa0

��ma0�

��s��

72�3 L�1Z �s0

0se�s=M

2ds�

1

12�h�sG

2i�L�1 �

�s��9�

�muh �uui� �mdh �ddi��L�1

�2�s��

9��muh �ddi� �mdh �uui��L

�1

�8

3� �E � ln

�2

M2 � Ei��s0

M2

��

�1

6M2 �muh �dgs�Gdi� �mdh �ugs�Gui��L�14�3b�=3b �16

27

��sM2 �h �uui

2� � h �ddi2��L�8=b

�16

9

��sM2 h �uui�h

�ddi�L�8=b�L�2b��J�=2b: (12)

Note that the RG improvement for QCD sum rule results is very important. For instance, L�2b��J�=2b � 1 (or L�b��J�=b � 1)atM2 � 1 GeV2, while the value becomes 0.93 (or 1.08) atM2 � 2 GeV2. In calculating the mass sum rule for the lowest-

INDICATIONS OF THE POSSIBLE OBSERVATION OF . . . PHYSICAL REVIEW D 76, 094001 (2007)

094001-3

lying 1�� exotic state, in Eq. (9) we substitute f3;a0with

the expression given in Eq. (12), where use of fa0�

�0:380� 0:015� GeV and �s0 � �3:0� 0:2� GeV2 hasbeen made [23]. In the numerical analysis, we shall usethe following values at the scale � � 1 GeV:

h�sGa��Ga��i � �0:474� 0:120� GeV4=�4��;

h �uui h �ddi � ��0:24� 0:01�3 GeV3;

h �ugs�Gui h �dgs�Gdi � �0:8� 0:1�h �uui;

�mu �md�=2 � �5� 2� MeV: (13)

Consequently, the mass sum rule for the lowest-lying 1��

resonance can be obtained by taking the logarithm of bothsides of Eq. (9) and then applying the differential operatorM4@=@M2 to them. We choose the Borel window1 GeV2 <M2 < 2 GeV2, where the contribution originat-ing from higher resonances (and the continuum), which isdefined as

1�

R1s0dse�s=M

2Im�OPE�s�

1�

R10 dse

�s=M2Im�OPE�s�

; (14)

is less than 53% and the highest OPE term at the quark-gluon level is no more than 13%. Both are well undercontrol. To obtain a reliable estimate for the mass sumrule result, the contributions arising from higher reso-nances and the highest OPE term at the quark-gluon levelcannot be too large. The excited threshold s0 can bedetermined when the most stable plateau of the masssum rule result is obtained within the Borel window.Note that no stable plateau can be obtained for the masssum rule if the sum rule result does not contain the RGcorrections. (See also the discussions in Sec. IV.)

Numerically, we get the mass for the lowest-lying 1��

exotic meson:

m�1� �1:26� 0:15� GeV; (15)

corresponding to s0 � 2:5� 0:7 GeV2. The result for m�1

versusM2 is shown in Fig. 1, where the mass is very stablewithin the window. From Eq. (15), we know that the secondlowest-lying meson for states coupling to the operator Jshould not be a0�1450� since the resulting mass is lowerthan the experimental result of ma0�1450� � �1:474�0:019� GeV [1]. It should be stressed that the procedurefor performing the RG improvement on the ‘‘mass’’ sumrule is very important. If the anomalous dimension of thecurrent J was neglected, the stable sum rule could not beobtained within the Borel window and the resulting masswas reduced by 300 MeV.

We further study the existence of the 1�� nonet. Inevaluating the mass for the lowest-lying strange 1�� exoticmeson, we use ms�1 GeV� � �135� 15� MeV,h �sgs�Gsi=h �uui � h �sgs�Gsi=h �uui � 0:85� 0:05 as addi-tional inputs. Because it is still questionable whether the �800� exists, we therefore consider the following two

possible scenarios. In scenario 1, �800� is treated as thelowest-lying strange scalar meson with mass 0:8�0:1 GeV and f � 0:37� 0:02 GeV, which correspondsto �s0 � 2:9� 0:2 GeV2, while in scenario 2, K�0�1430� isconsidered as the lowest-lying strange scalar meson withfK�0�1430� � 0:37� 0:02 GeV, which corresponds to �s0 �

3:6� 0:3 GeV2. (The values are updated from Ref. [23].)The results are depicted in Fig. 2. Using an arbitrary set ofallowed inputs, within the Borel window, the mass is stableonly for scenario 2, which hints that the may not be a realparticle or suitable in the sum rule study due to its largewidth. Because it is not stable in scenario 1, we assumes0 � 2:6� 0:7 GeV2, consistent with the case of �1. Theresult in scenario 2 is

mK��1�� 1:31� 0:19 GeV; (16)

corresponding to s0 � 2:3� 0:9 GeV2.

1. 1.2 1.4 1.6 1.8 2.

M2 GeV2

0.75

1.

1.25

1.5

1.75

m1

VeG

FIG. 1 (color online). The mass for the lowest-lying 1��

exotic meson as a function of the Borel mass squared M2. Thesolid curve is obtained by using the central values of inputparameters. The region between the two dashed lines is variationof the mass within the allowed range of input parameters.

1. 1.2 1.4 1.6 1.8 2.

M2 GeV2

0.75

1.

1.25

1.5

1.75

mK

1Ve

G

FIG. 2 (color online). The same as Fig. 1 but for the mass ofthe lowest-lying strange 1�� exotic meson. The light (blue) andheavy (red) curves are for scenarios 1 and 2, respectively, wheres0 � 2:3� 0:9 �2:7� 0:7� GeV2.


094001-4

IV. THE MASS SUM RULE, AS DERIVED FROMTHE CORRELATION FUNCTION GIVEN IN THE

LITERATURE, REVISITED

Before concluding this paper, to clarify why our resultgiven in the previous section differs from those given in theliterature, where a larger mass ( * 1:55 GeV) for thelowest-lying 1�� hybrid meson was obtained, we takeinto account the following correction function:

iZd4xeiqxh0jJ��x�J

y� �0�j0i � �q�q� � q2g��v�q2�

� q�q��s�q2�; (17)

with

J��x� � i �d�x��gsG��x�u�x�; (18)

which was adopted in Refs. [13,14]. The resulting sum rulecan be written in the following form:

e�m2�1=M2

m4�1�f4;�1

�2 �1

�

Z s0

0dse�s=M

2Im�OPE

v �s�; (19)

where

h0jJ��0�j�1�p; �i � f4;�1m2�1"�� : (20)

The f4;�1determines the normalization of twist-4 LCDAs

of the lowest-lying 1�� meson. The detailed OPE result for�v can be found in Refs. [13,14], where the radiativecorrections have been calculated.

The mass can then be obtained by applying�M4@=@M2 ln� to both sides of Eq. (19). Before proceed-ing, three remarks are in order. First, although the anoma-lous dimension of J� was computed in Ref. [13], such acorrection was not taken into account in Refs. [13,14]. Thescale dependence of the operator J� is given by J��M� �J��L

32=�9b� [13]. Second, the scale dependence of thestrong-coupling constant is ignored in Refs. [13,14]. Third,for theoretical completeness, the RG effect should be in-cluded in the mass sum rule (19) as done in Eq. (9), so thatthe right-hand side of Eq. (19) should be multiplied by theoverall factor L1�64=�9b�. Such a factor is equal to 1 atM2 � 1 GeV2, but becomes �0:95 at M2 � 2 GeV2.

In Fig. 3, we plot the mass sum rule as a function of theBorel mass squared within 1 GeV2 <M2 < 2 GeV2. Thelight curves (the upper set) are the results,2 where wesimply fix the scale at 2 GeV for input parameters and

use s0 � 4 GeV2, which consist with that obtained by Jinand Korner [14] (see Fig. 3 in Ref. [14]). Moreover, if RGeffects are not considered, as in Ref. [14], there is noplateau for any value of s0. Nevertheless, the stable plateaucan be reached for a typical value of s0 if we take intoaccount the RG corrections in the sum rule [see the heavycurves (the lower set) in Fig. 3]. The corresponding resultsare

m�1� �1:18� 0:09� GeV; s0 � �2:4

�0:4�0:2� GeV2; (21)

where the errors are due to variation of the input parame-ters. Note that the plateau covers the range for M2 �1:5 GeV2. Unfortunately, for 1 GeV2 <M2 < 2 GeV2

and s0 � 2:4 GeV2, the contribution originating fromhigher resonances (and the continuum) lies between 72%and 95%, which is too large, so that the resulting mass isless reliable.

V. CONCLUSIONS AND DISCUSSIONS

In this work we have studied the mass of the lowest-lying 1�� hybrid meson using the QCD sum rule tech-nique. Instead of J��x� � i �d�x��gsG�

��x�u�x�, which isalways adopted in the literature, we use, for the first time,

J�x� � z�z� �d�x��gsG��x�u�x�;

as the interpolating current of the lowest-lying 1�� hybridmeson. Our main results are as follows:

(i) Using J as the interpolating current of the 1�� exoticmesons, which are the so-called 1�� hybrid mesons(see the discussions in Sec. II), we have obtained themass for the lowest-lying 1�� exotic meson to be�1:26� 0:15� GeV. The threshold of excited states,s0, as well as the resulting mass can thus be deter-mined due to the existence of the most stable plateau

1. 1.2 1.4 1.6 1.8 2.

M2 GeV2

0.75

1.

1.25

1.5

1.75

2.

m1

VeG

FIG. 3 (color online). The same as Fig. 1 but for the sum rulegiven in Eq. (19). The heavy (red) curves are obtained from theRG-improved sum rule, where the solid curve corresponds tos0 � 2:4 GeV2. The light (blue) curves, using s0 � 4:0 GeV2

and the scale � � 2 GeV for �s and the remaining parameters,do not contain RG corrections.

2In Ref. [13], Chetyrkin and Narison use the finite energy sumrule approach in the analysis and argue that s0 may lie between3:5 GeV2 and 4:5 GeV2. They obtain the mass of the lowest-lying 1�� hybrid meson to be 1:6� 1:7 GeV. In their analysis,the parameter used, 1=�, roughly agrees with M2 ��1:6� 2� GeV2. We thus find that the light curves in Fig. 3are consistent with the result obtained by Chetyrkin and Narison.


094001-5

of the mass sum rule result within the Borel window,where the contributions arising from the resonancesand the highest OPE term at the quark-gluon levelare well under control. The plateau for the massresult versus the Borel mass squared exists onlywhen the RG corrections are taken into account inthe sum rule. Our result is in good agreement withthe world average of data for the mass of �1�1400�,which is m�1�1400� � �1:376� 0:019� GeV.

(ii) On the other hand, to clarify the discrepancy be-tween our result and those given in the literaturefor which J� is used as the interpolating field ofthe 1�� hybrid meson, we have reexamined theprevious sum rule calculation. (See Sec. IV.) As inRef. [14], the curve for the resulting mass versus theBorel mass squared has no plateau region for anyvalue of s0 if RG corrections are not considered.Unfortunately, with RG corrections, the plateau cov-ers the range for M2 � 1:5 GeV2, where the contri-bution arising from higher resonances is * 80% inthe sum rule result, so that the resulting mass is lessreliable.

(iii) Using J in the sum rule study, we also obtain themass for the strange 1�� hybrid meson: mK��1��

1:31� 0:19 GeV. In the Particle Data Book (PDG)[1], the two strange mesons,K��1410� andK��1680�,are currently assigned to be 23S1 and 13D1 states,respectively. However, K��1410� is too light as com-pared with the remaining 23S1 nonet states, andtherefore it could be replaced by K��1680� as the23S1 state. If so, our result hints thatK��1410� is verylikely to belong to the lowest-lying 1�� nonet.

(iv) We have presented a discussion for properties of the1�� exotic meson (which is usually called the 1��

hybrid meson) based on the QCD field theory.Although q- �q-g Fock states of the exotic meson aresuppressed (see also discussions in Sec. I), the twist-2 distribution amplitudes, which are antisymmetricunder the interchange of the quark and antiquarkmomentum fractions in the SU(3) limit, give themain contributions to the deep exclusive electropro-duction involving a lowest-lying 1�� exotic meson.Thus, from the point of view of QCD, the crosssection of the above reaction is still sizable and theexotic (hybrid) meson can be studied in experimentsat JLAB, HERMES, or Compass [17]. Furthermore,because of the nonsmall first Gegenbauer moment ofthe twist-2 distribution amplitudes, the branchingratios for two-body B decays involving �1�1400�[24] could be of order 10�6, which are easily acces-sible at the B factories.

In conclusion, we may have evidence for the existenceof 1�� exotic mesons. These states are allowed in quantumchromodynamics but not in the conventional quark model.Theorists tried to predict these states, especially with mass& 1:6 GeV, but did not succeed. In this paper, we haveshown that the �1�1400� state is indeed expected in QCD.We also predict the mass of the lowest-lying strange 1��

exotic meson which is a little larger than m�1�1400�. Thus,the observation of the �1�1400� state can be direct evi-dence of QCD in explaining the confinement mechanism ofstrong interactions.

ACKNOWLEDGMENTS

This work was supported in part by the National ScienceCouncil of the Republic of China under Grant No. NSC95-2112-M-033-001.

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