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arXiv:0801.3635v3 [nucl-th] 4 Nov 2008

X(2175) as a resonant state of the φK¯K system

A. Mart´ ınez Torres1, K. P. Khemchandani1∗, L. S. Geng1,

M. Napsuciale1,3, and E. Oset1

1Departamento de F´ ısica Te´ orica and IFIC, Centro Mixto Universidad de Valencia-CSIC,

Institutos de Investigaci´ on de Paterna, Aptdo. 22085, 46071 Valencia, Spain

2Instituto de F´ ısica, Universidad de Guanajuato, Lomas del Bosque 103, Fraccionamiento Lomas

del Campestre, 37150, Le´ on, Guanajuato, M´ exico.

Abstract

We perform a Faddeev calculation for the three mesons system, φK¯K, taking the interac-

tion between two pseudoscalar mesons and between a vector and a pseudoscalar meson from

the chiral unitary approach. We obtain a neat resonance peak around a total mass of 2150

MeV and an invariant mass for the K¯K system around 970 MeV, very close to the f0(980)

mass. The state appears in I=0 and qualifies as a φf0(980) resonance. We enlarge the space

of states including φππ, since ππ and K¯K build up the f0(980), and find moderate changes

that serve to quantify theoretical uncertainties. No state is seen in I=1. This finding provides

a natural explanation for the recent state found at BABAR and BES, the X(2175), which

decays into φf0(980).

1Introduction

The discovery of the X(2175) 1−−resonance in e+e−→ φf0(980) with initial state radiation at

BABAR [1, 2], also confirmed at BES in J/Ψ → ηφf0(980) [3], has stimulated research around

its nontrivial nature in terms of quark components.

s¯ ss¯ s is investigated within QCD sum rules in [4], and as a gluon hybrid s¯ sg state has been

discussed in [5, 6]. A recent review on this issue can be seen in [7], where the basic problem

of the expected large decay widths into two mesons of the states of these models, contrary to

The possibility of it being a tetraquark

∗Present address: Centro de F´ ısica Computacional, Deptartamento de F´ ısica, Universidade de Coimbra, P-3004-

516 Coimbra, Portugal

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what is experimentally observed, is discussed. The basic data on this resonance from [1, 2] are

MX= 2175 ± 10 MeV and Γ = 58 ± 16 ± 20 MeV, which are consistent with the numbers quoted

from BES MX= 2186 ± 10 ± 6 MeV and Γ = 65 ± 25 ± 17 MeV. In Ref. [2] an indication of this

resonance is seen as an increase of the K+K−K+K−cross section around 2150 MeV. A detailed

theoretical study of the e+e−→ φf0(980) reaction was done in Ref. [8] by means of loop diagrams

involving kaons and K∗, using chiral amplitudes for the K¯K → ππ channel which contains the

f0(980) pole generated dynamically by the theory. The study revealed that the loop mechanisms

reproduced the background but failed to produce the peak around 2175 MeV, thus reinforcing the

claims for a new resonance around this mass.

In the present paper, we advocate a very different picture for the X(2175) resonance which

allows for a reliable calculation and leads naturally to a very narrow width and no coupling to two

pseudoscalar mesons. The picture is that the X(2175) is an ordinary resonant state of φf0(980)

due to the interaction of these components. The f0(980) resonance is dynamically generated from

the interaction of ππ and K¯K treated as coupled channels within the chiral unitary approach of

[9, 10, 11], qualifying as a kind of molecule with ππ and K¯K as its components, with a large

coupling to K¯K and a weaker one to ππ [hence, the small width compared to that of the σ(600)].

Similar studies for the vector-pseudoscalar interaction have also been carried out using chiral

dynamics in [12, 13], which lead to the dynamical generation of the low-lying axial vectors. We

shall follow the approach of Ref. [13] to deal with this part of the problem and will use the φK

and φπ amplitudes obtained in that approach.

To study the φf0(980) interaction, we are thus forced to investigate the three-body system

φK¯K considering the interaction of the three components among themselves and keeping in mind

the expected strong correlations of the K¯K system to make the f0(980) resonance. For this purpose

we have solved the Faddeev equations with coupled channels φK+K−and φK0 ¯

later complemented with the addition of the φππ state as a coupled channel. The study benefits

from previous ones on the π¯KN and ππN along with their coupled channels done in [14, 15], where

many 1/2+, strange, and nonstrange low-lying baryon resonances of the Particle Data Group [16]

were reproduced. This success encourages us to extend the model of Refs. [14, 15] to study the

three-meson system, i.e., φK¯K. One of the interesting findings of Refs. [14, 15] was a cancellation

of the off-shell part of the amplitudes with the genuine three-body forces that one obtains from

the same chiral Lagrangians. This simplified technically the approach, and we shall stick to this

formalism also here.

K0. The picture is

2 Formalism

To study the φK¯K system, it is required to solve the Faddeev equations. The procedure followed is

(1) we solve coupled-channel Bethe-Salpeter equations for pseudoscalar - pseudoscalar meson (PP)

interaction as done in [9]; and for pseudoscalar-vector mesons (PV) interaction as in [13]; (2) then

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we solve the Faddeev equations for the three-body, i.e., vector-pseudoscalar-pseudoscalar (VPP)

mesons, system using the model developed in [14]. We describe the input and the formalism for

this latter part briefly in this section.

We calculate the three-body T matrix as obtained in Ref. [14] in terms of TR, i.e.,

TR= T12

R+ T13

R+ T21

R+ T23

R+ T31

R+ T32

R, (1)

where

Tij

R= tigijtj+ ti?

GijiTji

R+ GijkTjk

R

?

i ?= j ?= k = 1,2,3 (2)

corresponds to the sum of all of the diagrams with the last two t matrices being tjand ti. The

ti(tj) in Eq. (2) denotes the two particle scattering matrix where the particle i(j) is a spectator.

In the chiral formalism of [9, 13], these t-matrices in L = 0, which we consider here, depend on the

total energy in the center of mass of the interacting pair. The Tij

partitions, Ti, as

Ti= tiδ3(?k′

Rcan be related to the Faddeev

i−?ki) + Tij

R+ Tik

R

(3)

where Tisums all of the diagrams with the particle i as a spectator in the last interaction and?ki

(?k′

The propagator gijin Eq. (2) can be expressed as

i) is the initial (final) momentum of the ith particle in the global center of mass.

gij=

?

D

?

r=1

1

2Er

?

1

√s − Ei(?ki) − Ej(?kj) − Ek(?ki+?kj) + iǫ

(4)

where√s is the total energy in the global CM system, El=

l, and D is the number of particles propagating between two consecutive interactions. The model

in Ref. [14] has been built by writing the terms including more than two t-matrices by replacing

the “gij” propagator by a function Gij k, thus leading to Eq. (2). The function Gij kis given by

??k2

l+ m2

lis the energy of the particle

Gij k=

?

d3k′′

(2π)3

1

2El

1

2Em

Fij k(√s,?k′′)

√slm− El(?k′′) − Em(?k′′) + iǫ,

(5)

where i ?= j, j ?= k, i ?= l ?= m,√slmis the invariant mass of the (lm) pair, and Fij kis defined as

Fij k= tj(√sint(?k′′))

gjk|on−shell

This Gij kis a loop function of a propagator, in the three-body scattering diagrams, in which the

dependence on the loop variable of an anterior t matrix and propagator has been included in the

form of an off-shell factor Fijk. This simplifies technically solving Eq. (1) and induces regrouping

?gjk|off−shell

?

[tj(√sint(?kj′))]−1.(6)

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of the three-body diagrams giving six Faddeev partitions [Eq. (2)] instead of three (see [14] for a

more detailed discussion).

We label φ as particle 1 and K and¯K as particle 2 and 3, respectively. The invariant mass of

the K¯K system√s23is taken as an input to the three-body calculations and is varied around the

mass of the f0. The K¯K interaction t1in this region contains the pole of the f0(980) [9, 10]. The

other invariant masses s12and s13can be then calculated in terms of the√s23and total energy

[14]. Thus, there are two variables of the calculations, i.e., the total energy and the invariant mass

of the K¯K system.

We shall now discuss the input, i.e., the two-body t matrices for the PP and PV mesons

interaction. For the PP case, the Bethe-Salpeter equation

t = V + V˜Gt(7)

has been solved for five coupled channels, i.e., K+K−, K0¯K0, π+π−, π0π0, and π0η. The potentials

V are calculated from the lowest order chiral Lagrangian and the loops˜G have been calculated

using dimensional regularization as in [9]. The authors of [9, 10, 11] found poles in the t matrices, in

the isospin 0 sector, corresponding to the σ and the f0resonances, and also the one corresponding

to the a0(980) for the isospin 1 case. It was also found that the f0resonance is dominated by the

K¯K channel and the pole for the f0appears at ∼ 973 MeV even when the ππ channel is eliminated.

The matrix element corresponding to the K¯K → K¯K scattering is used as an input, t1, to solve

Eqs.(2) and (1). In the two-body problem, the f0(980) pole appears below the K¯K threshold. It

corresponds to total energies of the K¯K system below 2mkand in the momentum representation to

purely imaginary kaon momenta if we take p2

avoid using unphysical complex momenta in the three-body system, we give a minimum value of

about 50 MeV/c to the kaon momentum in the K¯K center of mass system. It should be mentioned

that the results are almost insensitive to this choice of the minimum momentum. For example, a

change in this momentum by about 40% changes the position of the peak merely by ∼ 5 MeV.

For the VP meson interaction, Eq. (7) is calculated with φK, ωK, ρK, K∗η, and K∗π as

coupled channels. The potential for the VP meson-meson interaction has been obtained from the

lowest order chiral Lagrangian and projected in the s wave [13], and then the φK → φK element

of the resulting coupled-channel t matrix is used as an input in Eq. (2)

Coming back to the three-body problem, we take the φ¯K(K) → φ¯K(K) t-matrix element as

t2(t3) to solve Eqs.(2) and (1). Our interest is to check the possibility of existence of a resonance

or a bound state with isospin zero in the φK¯K system; thus the full TRmatrix [Eq. (1)] is to be

projected to total isospin 0. When adding the φππ channel, we must deal with the ππ and φπ

interactions which are part of the coupled-channel study of the scalar and axial vector resonances,

respectively.

K= m2

K(which is not the case in a bound state). To

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3A discussion on possible coupled channels

In the construction of the K¯K and φK two-body t matrices we have used the full space of coupled

channels as indicated in Sec. 2. We shall argue here that in the three-body basis we can omit some

states. The φK system couples to ωK, ρK, K∗π, and K∗η. We shall bear in mind that we are

looking for a state with I = 0 and with√s23≃ 980 MeV, as found in the experiment [1, 2]. When

adding the¯K of the three-body φK¯K system to the coupled channels of the φK, we obtain the

following states: ωK¯K, ρK¯K, K∗π¯K, and K∗η¯K. If we want the subsystem of two pseudoscalar

mesons to build up the f0(980), which is dynamically generated in the K¯K and ππ interaction,

we must exclude the K∗π¯K and K∗η¯K states. The ρK¯K state is also excluded because when K¯K

couples to the f0(980) the total isospin of the state is I = 1. Only the ωK¯K state is left over. We

could add this channel to the φK¯K, but the ωK¯K channel lies ∼ 400 MeV below the X (2175)

resonance mass and hence is not expected to have much influence in that region. In more technical

words, a channel which lies far away from the energy region under investigation would only bring

a small and smooth energy-independent contribution to the final amplitude because of the large

off-shellness of the propagators.

Thus the introduction of the ωK¯K channel can only influence mildly the results obtained with

the φK¯K system alone, and thus we neglect it in the study. Furthermore we have also seen that

the φK → ωK and ωK → ωK amplitudes are weaker than the φK → φK one.

Even though we argue above that¯K∗πK and¯K∗ηK channels should be neglected, we have also

investigated the effect of including the¯K∗πK channel, as an example. This is a channel where the

πK interaction (together with the ηK channel) leads to the scalar κ resonance, and actually there

are works which hint towards a possibility of¯K∗κ forming a molecule with mass around 1576 MeV

[17]. What we find can be summarized as follows:

• In the energy region of our interest, we find a small transition amplitude from φK → K∗π

as compared to φK → φK, indicating a small mixture of the φK¯K, and¯K∗πK components.

• Studying the¯K∗πK system alone, we find that the corresponding amplitudes are much

smaller in size than those found in the φK¯K system in the energy region around 2150 MeV.

• In the region of energies around 1600 MeV, the¯K∗πK amplitudes can be bigger than around

2150 MeV, but they are still smaller than the φK¯K amplitude at 2150 MeV.

From these findings we conclude that, although more detailed work needs to be done at energies

around 1600 MeV to check the suggestion of [17], the amplitude of the¯K∗πK channel in this energy

region seems too weak to support bound states. On the other hand, we can be more assertive by

stating that the effect of the¯K∗πK channel around 2150 MeV is negligible.

We can now stick to having the φ as the vector meson and K¯K as the main meson-meson

channel. Yet, K¯K and ππ are strongly coupled in I=0, both the K¯K → K¯K and ππ → ππ

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amplitudes are strong, and it is only the intricate nonlinear dynamics of coupled channels of the

Bethe-Salpeter equations that produces at the end two states, the σ that couples strongly to the

ππ channel and the f0(980) that couple strongly to K¯K. Hence, we find advisable to include φππ

as a coupled channel.

4 Results

In Fig. 1, we show the squared amplitude | TR|2and its projection, as a function of the total

energy (√s) and the invariant mass of the K¯K system (√s23), in the isospin zero configuration.

We have made the isospin projection of the amplitude of Eq. (1) using the phase convention

| K−? = − | 1/2,−1/2? as

| φK¯K;I = 0,IK¯ K= 0? =

1

√2

?

| φK+K−?+ | φK0¯K0?

?

. (8)

A clear sharp peak of | TR|2can be seen at 2150 MeV, with a full width at half maximum ∼

16 MeV. In order to make a meaningful comparison of this width with the experimental results,

we have folded the theoretical distribution with the experimental resolution of about 10 MeV and

then we find an appropriate Breit-Wigner distribution with a width Γ ∼ 27 MeV. The peak in

| TR|2appears for the√s23∼ 970 MeV which is very close to the pole of the f0resonance [9].

960

970

980

√s23 (MeV)

2050

2100

2150

2200

2250

2300

√s (MeV)

0

20

40

60

80

|TR|2 (MeV-4)

Figure 1: The φK¯K squared amplitude in the isospin 0 configuration.

The total mass, the invariant mass of the K¯K subsystem and the quantum numbers IGJPC=

0−1−−of the resonance found here are all in agreement with those found experimentally for the

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X(2175) [1, 2]. These findings strongly suggest that this resonance can be identified with the

X(2175).

Yet, our approach can go further and we can make an evaluation of the production cross section

and compare it with the experimental results of [1, 2]. For this we make use of the theoretical

evaluation of the φf0(980) production in the e+e−reaction studied in [8]. The authors in [8] studied

the production of the φ and f0(980) as plane waves (pw) in the final state and could reproduce the

background but not the peak structure around X(2175) mass. Since our resonance develops from

the interaction of the φ and f0, the consideration of the final state interaction (fsi), in addition

to the uncorrelated φf0production amplitude (Tφf0

in the peak region. We show here that this is indeed the case. We implement the φf0 fsi by

multiplying Tφf0

pw) of [8], could explain the experimental data

pwby the factor [18, 19, 20, 21]

Ffsi= [1 +˜Gφf0(s)tφf0(s)], (9)

where tφf0is the scattering matrix for φ and f0and˜Gφf0(s) is the loop function of the φ and f0

propagators. For˜Gφf0we use the standard formula for two mesons [9] with a cut-off (Λ) of the

order of the sum of the two meson masses, as was the case in [9], and hence Λ ∼ 2 GeV here. We

do not have the tφf0, but in the vicinity of the resonance it must be proportional to the three-body

TR [Eq. (1)], implying Tφf0= αTR. The proportionality coefficient α is readily obtained using

a relation based on unitarity, Im{T−1

channel to be the main source of Im{TR}, as the experimental study suggests [1, 2], we have

φf0} = −Im{˜Gφf0}, implicit in Eq. (7). Assuming the φf0

Im{T−1

φf0} = α−1Im{T−1

R} = −Im{˜Gφf0} =

kφ

8π√s,

(10)

which determines α. In Eq. (10), kφis the φ momentum in the φf0center of mass system.

With this information we evaluate the e+e−→ φf0production cross section taking the results

for the φf0production in the plane wave approximation from [8], and we show the results in Fig.2.

We can see that taking a cut-off of the order of 2-2.5 GeV for the˜Gφf0, we obtain results for

the production cross section which are in fair agreement with the experimental ones. In order to

compare the results with the experimental cross sections in the X (2175) mass region, the energy

argument of the amplitude TRhas been shifted by ∼ 25 MeV. Note, however, that the difference of

25 MeV in the energy position (1 % of the mass) represents a remarkable agreement for a hadronic

model of meson spectra.

We would like now to comment on the effects of including the φππ channel, as discussed in Sec.

3 . We observe a similar peak as in Fig. 1 (see Fig. 3); however, the position of the peak in the

total energy has been displaced by about 38 MeV downwards to an energy of 2112 MeV. At the

same time, the peak shows up around√s23≃ 965 MeV, about 15 MeV below the nominal energy

of the f0 (980). These differences with the nominal values of the masses of the resonances are

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200022002400

2600

2800

s (MeV)

0.1

0.2

0.3

0.4

0.5

0.6

σ (φf0) (nb)

Figure 2: The cross section for the e+e−→ φf0reaction. The dashed-dotted line shows the result

of the calculation of the cross section in the plane wave approximation [8]. The dashed (solid) line

shows the result of multiplying the amplitude from Ref. [8] by the final state interaction factor

[Eq. (9)] calculated using a cutoff of 2 (2.5) GeV for the˜Gφf0(s). The data, which corresponds

to the e+e−→ φ(ππ)I=0reaction (triangles for charged pions and boxes for neutral pions), have

been taken from [1, 2].

typical of any hadronic model of resonances and, thus, the association of the resonance found to

the X (2175), which has the same quantum numbers as the resonance found, is the most reasonable

conclusion. In any case, the different options taken along the work, have always led to a clean

peak around the same position, and the difference found could give us an idea of the theoretical

uncertainties.

Finally, it should be mentioned that the TR matrix for the isospin = 1 does not show any

structure.

We have checked the sensitivity of the resonance found to the change in the cut-off (Λ ∼ 1000

MeV) used in the calculation of the input two-body t matrices [Eq. (7)], which gives the same

results as dimensional regularization. There is not much freedom to change the Λ in this case,

because it has been constrained by reproducing the data on the respective two-body scattering. We

thus change Λ by ∼ 20 MeV for calculating Eq. (7), which still guarantees a reasonable agreement

with the two-body cross sections, and find that it gives rise to a change in the peak position (in

Fig.1) in√s by ∼ 8 MeV. The cutoff is also needed to evaluate the G functions of Eq. (5),

and we use the same cut-off of about 1 GeV. Since this function involves loops with three-meson

propagators, it is very insensitive to the cutoff. The same change of ∼ 20 MeV (or more) in Λ

leads to negligible changes in the results in this case.

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960

965

970

√s23 (MeV)

2040

2080

2120

2160

√s (MeV)

0

20

40

60

80

|TR|2 (MeV-4)

Figure 3: The squared amplitude in the isospin 0 configuration including the φππ channel.

5Off-shell effects and three-body forces

Our approach makes explicit use of the cancellation of the off-shell parts of the two-body t matrices

in the three-body diagrams with the genuine three-body forces, which arise from the same chiral

Lagrangians . The off-shell part of a scattering matrix is unphysical and can be changed arbitrarily

by performing a unitary transformation of the fields.

Inside the loops, the off-shell part of the chiral amplitudes, which behaves as p2− m2(where

p is the four vector of the off-shell particle) for each of the meson legs, cancels a propagator

leading to a diagram with the topology of a three-body force [14]. It is also a peculiar feature of

the chiral Lagrangians that there is a cancellation of these three-body forces with those arising

from the PPV → PPV contact terms of the theory. Examples of similar cancellations are well

known in chiral theories [22, 23]. The detailed derivation of the cancellation of the off-shell part

of the t matrices and the three-body force arising from the chiral Lagrangian can be seen in the

appendix [15] for the ππN interaction. In the present case, the cancellation also occurs, but it

is slightly different technically. In fact, its derivation is easier than in the case studied in [15],

and we discuss it in the appendix for a case in which the leading order contribution to the V P

transition is not null. However, in the φK → φK case, the potential is zero. In this case, the t

matrix is generated by rescattering through K∗π and K∗η states, and the cancellation is found in

the transition potentials.

We find also instructive to see what one gets if the off-shell part of the two-body tmatrices is

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retained. Following Refs. [9, 13] we find, for the s wave,

VK¯ K(I = 0) = −1

VφK≃3

4f2(3s23−

?

?

i

(p2

i− m2

i))(11)

2s12−1

2

?m2

i−1

2

i

(p2

i− m2

i).(12)

The (p2

[9] but will show up in the external legs of the two-body t matrix used as an input in the Faddeev

equations. Hence

i−m2

i) terms in Eq. (11) are ineffective in the loops of the two-body t matrix [Eq. (7)]

tK¯ K(I = 0) = ton

?

1 −

?

i(p2

i− m2

3s23

i− m2

im2

i)

?

(13)

tφK= ton

?

1 −

?

3s12−?

i(p2

i)

i

?

, (14)

where tondenotes the corresponding on-shell t-matrix. If we use these amplitudes, instead of the

on-shell ones we find a very similar result to that depicted in Fig.1, with the amplitude peaking

at√s = 2110 MeV and√s23= 975 MeV. Thus, the K¯K still appears very correlated around the

f0(980), but the total energy has been shifted by 40 MeV. This is the result we obtain by using

the off-shell t matrices and neglecting the effect of the PPV → PPV contact term of the theory,

which as mentioned above cancels the effect of the off-shell part of the t matrix. In other words,

we could say that the three-body forces of the chiral Lagrangian are responsible for a shift of the

resonance mass from 2110 to ∼ 2150 MeV, thus leading to a better agreement with the mass of the

X(2175), but, of course, the result holds for the particular choice of fields of the ordinary chiral

Lagrangians.

6Pole in the complex plane

One might want to see if a peak obtained in the three-body T matrix corresponds to a pole in the

complex plane. The peak is so clean and close to a Breit-Wigner for a fixed√s23that it can only

be reflected by a pole in the complex plane. Yet, we have looked at it in more detail through in a

simplified way. We keep the variable√s23as real, and we fix its value to the one where the peak

appears and then study the φK¯K amplitude as a function of the complex√s variable. We must

move to the second Riemann sheet in the φf0(980) amplitude which is accomplished by changing

kφ→ −kφin the φf0loop function. We proceed as explained below.

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The unitarity condition allows us to write [24]

T−1

φf0= V−1

φf0−˜Gφf0

(15)

with Vφf0the real potential and˜Gφf0the φf0loop function used in Eq. (9)

Going to the second Riemann sheet implies substituting˜Gφf0by˜GII

changing kφby −kφin the analytical expression of˜Gφf0[13]

˜Gφf0(√s) =

16π2

[ln(s − (m2

ln(s − (m2

φf0, where˜GII

φf0is obtained

1

?

a(µ) + lnm2

φ

µ2+m2

√s) + ln(s + (m2

√s) − ln(s + (m2

f0− m2

2s

φ+ s

lnm2

f0

m2

φ

+kφ

√s

√s) −

√s) − 2πi]

(16)

φ− m2

f0) + 2kφ

φ− m2

f0) + 2kφ

φ− m2

f0) − 2kφ

φ− m2

f0) − 2kφ

?

Thus we can write

(T−1

φf0)II

= (V−1

= (T−1

φf0) − (˜Gφf0)II

φf0)I+˜Gφf0− (˜Gφf0)II

= (T−1

(17)

φf0)I− i

kφ

4π√s,

where I and II indicate the first and second Riemann sheet, respectively. We can approximate TR

of Eq. (2) by a Breit-Wigner form as

TR≃

g2

s − so+ iMΓ(s)

(18)

from where, by means of Eq. (10), since α is real

(T−1

φf0)I= (α−1T−1

R)I=

?

kφ

8π√sIm{T−1

R}

?

√s=MT−1

R

(19)

which leads to

(T−1

φf0)II=

?

kφ

8π√s

1

MΓ

?

√s=M(s − so+ iMΓ) − i

kφ

4π√s,

(20)

which upon taking into account that

Γ =

1

8πsg2kφ,(21)

with kφbeing real, results in

(T−1

φf0)II=1

g2

?

s − s0− ikφ

8π

?2√s − M

s

?

g2

?

.(22)

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Then (T−1

φf0)IIhas a pole at

s − s0− ikφ

8π

?2√s − M

s

?

g2= 0. (23)

which appears indeed very close to Re√s ≃√s0and Im√s ≃ Γ/2 as we have checked numerically,

taking s0and g2from the shape of TR. We also get the complex conjugate pole taking another

branch of the logarithm.

7 Summary

In summary, the interaction of the φK¯K system studied with the Faddeev equations leads to a

rearrangement of the K¯K subsystem as the f0(980)(0++) resonance. Then, the f0(980) together

with the φ forms a narrow resonant 1−−state with a mass bigger than mφ+2mK, which decays into

φf0(980) and hence is most naturally associated to the recently discovered X(2175) resonance. The

narrow width of around 27 MeV obtained here is compatible within errors with the experimental

width 58±16±20 MeV. We have also included φππ as a coupled channel of φK¯K and find a peak

very similar to the one found with the φK¯K channel alone, except that the peak is displaced by

38 MeV down to smaller masses. We also noted that the theoretical uncertainties are of this order

of magnitude.

The typical differences of our results with the experimental ones are in the range of 50 MeV for

the mass and the width are roughly compatible. These are typical differences found in successful

models of hadron spectroscopy. The theory also shows that there is no resonance in φa0(980).

Although a complete study of this state would require the addition of the φηπ channel, we found

that the strength of the φK¯K amplitude in I = 1 is much smaller in magnitude than that of the

φK¯K in I = 0, far away from developing a pole upon reasonable changes of the input variables.

It would be most interesting to test experimentally this prediction.

8 Acknowledgement

This work is partly supported by DGICYT Contract No. FIS2006-03438 and the Generalitat

Valenciana. A. M. T. acknowledges support from the Ministerio de Educaci´ on y Ciencia. M. N.

acknowledges support from CONACyT-M´ exico under Project No. 50471-F. K.P.K thanks the EU

Integrated Infrastructure Initiative Hadron Physics Project under Contract No. RII3-CT-2004-

506078 and the Funda¸ c˜ ao para a Ciˆ encia e a Tecnologia of the Minist´ erio da Ciˆ encia, Tecnologia

e Ensino Superior of Portugal for financial support under Contract No. SFRH/BPD/40309/2007.

12

Page 13

Appendix

In [15], a cancellation between the three-body force whose origin is in the off-shell part of

the t matrices and the one arising from the chiral Lagrangian was shown for the ππN system

(as an example of a two-meson–one-baryon system). Here we are going to show that the same

cancellation also occurs in the case of one vector and two pseudoscalar mesons. In this article, we

have considered φππ and φK¯K as the main coupled channels. However, the φπ and φK (¯K) t

matrices have been calculated taking the coupled channels of [13], since the potentials for φπ → φπ

and φK(¯K) → φK(¯K) are zero. This means that the φπ and φK(¯K) interactions proceed through

other coupled channels. Therefore, in order to see the mentioned cancellation for the φππ and φK¯K

channels, we must consider at least one loop for the φπ → φπ and φK(¯K) → φK(¯K) interaction in

which the intermediate state is one of the coupled channels of [13]. The cancellations in this case

have to be seen in the terms involving the transition to these intermediate states of the coupled

channels. This can be done in the same way as it will be shown below, but for the sake of clarity we

have taken a simple case to show the cancellation between the contribution of the off-shell part of

the t matrices and the contact term vector-pseudoscalar-pseudoscalar of the corresponding chiral

Lagrangian. We consider the channel ρ+π+π−, for which the ρ+π+(π−) → ρ+π+(π−) transition is

not zero at leading order, as an example. In order to simplify the formulation we take ρ+π+π−as

the only channel of the system.

The interaction of a vector and any number of pseudoscalar mesons is described by the chiral

Lagrangian [13]

L = −Tr{[Vµ,∂νVµ]Γν}

where

(24)

Vµ =

1

2(u†∂νu + u∂νu†)

= ei

1

√2ρ0+

1

√2wρ+

K∗+

K∗0

φ

ρ−

K∗−

−1

√2ρ0+

¯K∗0

1

√2w

µ

Γν

=

u2

√2

fP

P=

1

√2π0+

1

√2η8

π+

K+

K0

−2

π−

K−

−1

√2π0+

1

√6η8

¯K0

√6η8

If we expand u in series up to terms containing two pseudoscalar fields P, we obtain

Γν=

1

4f2[P,∂νP] (25)

13

Page 14

k1

k′

1

ρ+

ρ+

π+

k2

k′

2

π+

Figure 4: Lowest order diagram contributing to the ρ+π+interaction.

and Eq. (24) becomes

LV P= −

1

4f2Tr{[Vµ,∂νVµ][P,∂νP]} (26)

For the case under consideration, i.e., ρ+π+→ ρ+π+and ρ+π−→ ρ+π−, Eq. (26) has the

form

L = −

2f2

1

?

∂µρ−

νρ+ν− ρ−

ν∂µρ+ν??

∂µπ−π+− π−∂µπ+?

(27)

leading to (see Fig. (4))

Vρ+π+→ρ+π+

Vρ+π−→ρ+π−

= −1

= −Vρ+π+→ρ+π+

2f2(k1+ k1′)(k2+ k2′)(ǫ · ǫ′)

(28)

From [15], we have

Vπ+π−→π+π− = −1

6f2

?

3sππ−

?

i

(k2

i− m2

i)

?

(29)

where kiand mirepresent the momentum and mass, respectively, of the external particles for the

π+π−interaction.

In this way, the contribution of the first diagram in Fig. (5) is given by

Ta = −

= Ton

1

6f2

a + Toff

?

3(k′

2+ k′

3)2− (k′2− m2

π)

?

1

k′2− mπ2

?

−

1

2f2(k1+ k′

1)(2k2+ k′− k2)

?

(ǫ · ǫ′)

a

,(30)

with Ton

matrices:

a

(Toff

a

) being the contribution which comes from the on-shell (off-shell) part of the t

Ton

a

=

1

2f4(k′

2+ k′

3)2

1

(k1+ k2− k′

1)2− m2

π

(k1+ k′

1)k2(ǫ · ǫ′)

14

Page 15

ρ+

π+

π−

ρ+

π+

π−

k1

k2

k3

k′

k′

k′

1

2

3

k′

ρ+

π+

π−

ρ+

π+

π−

k1

k2

k3

k′

k′

k′

1

2

3

k′

ρ+

π+

π−

ρ+

π+

π−

k1

k2

k3

k′

k′

k′

1

2

3

k′

ρ+

π+

π−

ρ+

π+

π−

k1

k2

k3

k′

k′

k′

1

2

3

k′

ρ+

π+

π−

ρ+

π+

π−

k1

k2

k3

k′

k′

k′

1

2

3

k′

ρ+

π+

π−

ρ+

π+

π−

k1

k2

k3

k′

k′

k′

1

2

3

k′

(a)(b)(c)

(d)

(e)(f)

Figure 5: Diagrams in which the off-shell part of the t matrices lead to a three-body force.

Toff

a

=

?

1

4f4(k′

2+ k′

3)2(k1+ k′

1)

?

k′− k2

k′2− m2

π

?

k′=k1+k2−k′

1

−

1

12f4

k′2− m2

k′2− m2

π

π

(k1+ k′

1)(k2+ k′)k′=k1+k2−k′

1

?

(ǫ · ǫ′)(31)

In analogy with the findings of [15], the contribution of the off-shell part for the different

diagrams of Fig. (5), together with one of the vector-pseudoscalar-pseudoscalar contact terms of

the chiral Lagrangian, is expected to vanish in the limit of equal masses for the pseudoscalars and

equal masses for the vectors. From Eq. (30) and following [15],

Toff

a

=

?

1

4f4(k′

2+ k′

3)2(k1+ k′

1)

∆k1

(∆k1)2+ 2k2∆k1

−

1

12f4(k1+ k′

1)(2k2+ ∆k1)

?

(ǫ · ǫ′)(32)

with ∆k1= k1− k′

1. Using that

(k1+ k′

1)∆k1 = k2

1− k′

1

2= m2

1− m′

1

2

(33)

is zero in the limit of equal masses, we have

Toff

a

= −

1

6f4(k1+ k′

1)k2(ǫ · ǫ′) (34)

By analogy, for the rest of the diagrams in Fig. (5) we have

Toff

b

=

1

6f4(k1+ k′

1)k3(ǫ · ǫ′)

15

Page 16

k1

k′

1

ρ+

π+

π−

ρ+

π+

π−

k2

k3

k′

2

k′

3

Figure 6: Contact term whose origin is in the chiral Lagrangian.

Toff

c

=

1

6f4(k1+ k′

1

6f4(k1+ k′

= 0

= 0

1)k′

3(ǫ · ǫ′)

Toff

d

= −

1)k′

2(ǫ · ǫ′)

Toff

e

Toff

f

Adding all Toffwe get

i=6

?

i=1

Toff

i

=

1

6f4(k1+ k′

1)(k′

3− k2+ k3− k′

2)(ǫ · ǫ′) (35)

In order to evaluate the VPP contact term, we have to expand Γν up to terms with four

pseudoscalar fields

Γν=

1

32f4

?1

3∂νPP3− P∂νPP2+ P2∂νPP −1

3P3∂νP

?

(36)

and, therefore, using Eq. (24), the chiral Lagrangian for the VPP contact term for the ρ+π+π−

interaction is (Fig. (6))

LV PP= −

1

12f4(∂µρ−

νρ+ν− ρ−

ν∂µρ+ν)(π−π−π+∂µπ+− π−∂µπ−π+π+)(37)

which implies

T3b

ρ+π+π− = −1

6f4(k1+ k′

1)(k′

3− k2+ k3− k′

2)(ǫ · ǫ′)(38)

The sum of Eq. (35) and Eq. (38) results in

6

?

i=1

Toff

i

+ T3b

ρ+π+π− = 0(39)

16

Page 17

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18

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