A Simple Explanation for the X(3872) Mass Shift Observed for Decay to D^{0} {D^0}bar

a r X i v :0710.5191v 2 [h e p -e x ] 13 D e c 2007

SLAC-PUB-12934

A Simple Explanation for the X (3872)Mass Shift Observed for Decay to D ?0ˉD

0W.Dunwoodie and V.Ziegler

Stanford Linear Accelerator Center,Stanford,California 94309,USA

(Dated:February 2,2008)

We propose a simple explanation for the increase of approximately 3MeV/c 2in the mass value of

the X (3872)obtained from D ?0ˉD

0decay relative to that obtained from decay to J/ψπ+π?.If the total width of the X (3872)is 2-3MeV,the peak position in the D ?0ˉD

0invariant mass distribution is sensitive to the ?nal state orbital angular momentum because of the proximity of the X (3872)to

D ?0ˉD

0threshold.We show that for total width 3MeV and one unit of orbital angular momentum,a mass shift ～3MeV/c 2is obtained;experimental mass resolution should slightly increase this value.A consequence is that spin-parity 2?is favored for the X (3872).

PACS numbers:14.40.Gx,13.25.Hw

The X (3872)has been seen primarily in its J/ψπ+π?decay mode [1,2,3,4,5],from which a measured mass value of 3871.4±0.6MeV/c 2is obtained [6];an upper limit on the total width of 2-3MeV has been estimated [1].Observation of decay to the J/ψγ?-nal state [7,8]has established positive C-parity,and analysis of the decay angular distributions [9]has nar-rowed the spin-parity (J P )possibilities to 1+or 2?for the X (3872).The invariant mass distribution for the

D 0ˉD

0π0system resulting from B-decay to the D 0ˉD 0π0K ?nal state shows a peak near threshold yielding a mass value 3875.2±0.7+0.9?1.8MeV/c 2[10],and this has been interpreted as evidence for the decay process

X (3872)→D ?0ˉD

0.A recent BaBar analysis [11]has con?rmed this result,and has obtained the correspond-ing mass value 3875.1+0.7?0.5±0.5MeV/c 2

.In each case,the ?rst error quoted is statistical,and the second is systematic.These results are in excellent https://www.wendangku.net/doc/117310586.html,pared to the mass value 3871.4±0.6MeV/c 2[6]for

the J/ψπ+π?decay mode,the mass di?erence is 3.8+1.2

?2.0

MeV/c 2from Belle,and 3.7+1.1?0.9MeV/c 2

from BaBar.The signi?cance of the latter is at the four standard devi-ation level,and given the consistency of the BELLE and

BaBar results for D ?0ˉD

0,it would seem to be a real e?ect.We take the point of view that this is indeed the case,and suggest a possible explanation,which,although very simple,carries some signi?cant physical implications.Consider the decay process B →KX ,with X is a resonance decaying to a ?nal state F .The invariant mass distribution for the system F takes the form d N

(m 20

?

m 2)

2

+

m 20Γtot (m )

2,(1)

where C 1is a constant,m is the invariant mass of system

F ,T B (m )is the invariant amplitude describing the B to KX coupling,T F (m )is the invariant amplitude describ-ing the X to F coupling,d φF (m )is the element of F de-cay phase space,and the denominator is the square of the relativistic Breit-Wigner propagator describing the reso-nance X .The factor m is present because the Lorentz-invariant phase space volume element is proportional to

d m 2,and (p/m B )is th

e two-body phase space factor for B →KX decay,with

p =

2m B

.

(2)

Equation (1)can be written in terms of the partial width for X decay to F ,ΓF (m ),as

d N

(m 20

?

m 2)

2

+m 20Γtot (m )

2.(3)

In general,Γtot (m ),which is the mass-dependent total

width of X ,takes the form

Γtot (m )=M i =1

Γi (m ),(4)

where M is the number of decay modes of X ,and the Γi

are the individual partial widths,of which ΓF is one.The amplitude T B is not known.However,if X has spin J ,angular momentum conservation requires that there be J units of orbital angular momentum associ-ated with the KX system resulting from B decay,and so we express T B in terms of the corresponding centrifugal barrier factor as follows:

T B (m )～

p J D J (p,R )

,

(5)

where D J (p,R )is the Blatt-Weisskopf Damping Fac-tor [12],and R is the associated radius parameter,for which we choose the value 5GeV ?1(i.e.1Fermi).The D J functions for J =0?3are summarized in Table I.For the X (3872),the mass range is limited (3.87-3.91GeV/c 2)and the Q-value for B →KX decay is large (0.9GeV/c 2),so that the m-dependence introduced by this description of the B decay vertex is small.

For the X (3872)only three decay modes have been ob-served to date [1,2,3,4,5,7,8,10,11],and of these the

01

11+(pR)2

29+3(pR)2+(pR)4

3225+45(pR)2+6(pR)4+(pR)6

[m2?(m D+m?

D )2][m2?(m D?m?

D

)2]

d m =C3 p2J+1/D J(p,R)

q2L+1/D L(q,R)

Γtot[MeV]

24

1.54 1.99

3871.4 1.17

0.45 1.02

X(3872)mass[MeV/c2]3

3870.8 4.25

2.37

3.61

3872.0 1.98

TABLE III:Dependence of the peak mass shift(in MeV/c2) on X(3872)mass and total width for L=1and J=2(i.e. J P=2?).

shift

in Fig.2(a)agrees well with the result from experi-ment[10,11],and the observed signal shapes seem better represented by that of Fig.2(b)than by that of Fig.1(b). However,it must be acknowledged that the experimen-tal uncertainties are signi?cant,and that even the uncer-tainty in the location of the D?0ˉD0threshold(above)is relevant on the scale of the e?ect under discussion.

FIG.1:(a)The m(D?0ˉD0)lineshape obtained from Eq.(11) for J P=1+using X(3872)mass3871.4MeV/c2(indicated by the dot-dashed line)and width3MeV.(b)The histogram for3000events generated using the curve shown in(a).

We have made no attempt to study the e?ect of de-tector resolution on the mass shifts calculated above.FIG.2:(a)The m(D?0ˉD0)lineshape obtained from Eq.(11) for J P=2?using X(3872)mass3871.4MeV/c2(indicated by the dot-dashed line)and width3MeV.(b)The histogram for3000events generated using the curve shown in(a). Near threshold,such e?ects should not be represented by Gaussian smearing in mass,as is done usually,since

this will yield contributions below threshold.It is three-momentum resolution which is the source of the smear-

ing,and this must be investigated by full detector sim-ulation for the experiment in question.Since such sim-ulation obviously cannot yield events below threshold,

it seems probable that the peak mass shifts calculated above will be increased as a result of experimental res-olution.We suspect that such e?ects will be small(<

1MeV/c2),but a thorough investigation making use of detector simulation is necessary.

In summary,we have shown that a simple treatment

of the orbital angular momentum involved in X(3872) decay to D?0ˉD0can account for the di?erence between the mass measured in this mode and that obtained from

J/ψπ+π?decay.The results favor J P=2?over J P= 1+for the X(3872),but the uncertainty in the measured mass di?erence(3.7±1.2MeV/c2),and the absence of simulated detector resolution e?ects,prevent a de?nite conclusion.If our interpretation is correct,a corollary is that the width of the X(3872)cannot be much smaller than～2MeV,since otherwise signi?cant displacement of the invariant mass peak for D?0ˉD0would not occur. Work supported by the U.S.Department of Energy under contract number DE-AC03-76SF00515.

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