\ phi K^{+} K^{-} production in electron-positron annihilation

21 Pages·2014·0.87 MB·English

φK+K− production in electron-positron annihilation.
S. G´omez-Avila1, M. Napsuciale1, E. Oset2
1 Instituto de F´ısica, Universidad de Guanajuato, Lomas del Bosque 103,
Fraccionamiento Lomas del Campestre, 37150, Le´on, Guanajuato, M´exico. and
2 Departamento de F´ısica Te´orica and Instituto de F´ısica Corpuscular,
Centro Mixto Universidad de Valencia-CSIC, 46000 Burjassot, Valencia, Spain.
In this work we study thee+e− →φ K+K− reaction. The leading order electromagnetic contri-
butions to this process involve the γ∗φ K+K− vertex function with a highly virtual photon. We
calculate thisfunction at low energies using RχPT supplemented with theanomalous term for the
VV′P interactions. Tree level contributions involve the kaon form factors and the K∗K transition
form factors. We improve this result, valid for low photon virtualities, replacing the lowest order
termsin thekaon form factors and K∗K transition form factors bytheform factors as obtained in
9 UχPT in the former case and the ones extracted from recent data on e+e− → KK∗ in the latter
0
case. We calculate rescattering effects which involve meson-meson amplitudes. The corresponding
0
resultisimprovedusingtheunitarizedmeson-mesonamplitudescontainingthescalarpolesinstead
2 of the lowest order terms. Using the BABAR value for BR(X →φf0)Γ(X →e+e−), we calculate

b the contribution from intermediate X(2175). A good description of data is obtained in the case of
e destructive interference between this contribution and the previous ones, but more accurate data
F on the isovector K∗K transition form factor is required in order to exclude contributions from an
intermediate isovector resonance to e+e− →φ K+K− around 2.2GeV.
7

PACSnumbers: 13.66.Bc,13.75.Lb,12.39.Fe.
]
h
p
- I. INTRODUCTION
p
e
h Recently, using the radiative return method [1, 2, 3], a new state, the X(2175) (also named Y(2175) in the
[ literature), was observed in e+e− φππ with the dipion invariant mass close to the f0(980) region, explicitly, for
→
2 mππ = 850 1100 MeV [2]. Later on this state was also detected at BES in the J/Ψ ηφf0(980) reaction [4] and
v BELLEin e−+e− φπ+π− [5]. In anupdate of the analysisof Ref. [2], results onthe ch→annele+e− φK+K− were
7 presentedinthef→ormofnumberofeventsasafunctionofthedikaoninvariantmass[3]. Indeed,inthi→swork,thecross
4 sectionfore+e− K+K−K+K− wasmeasuredasafunctionofthecenterofmassenergyupto√s=4.54MeV and
1 →
itisshowntherethatthisreactionisdominatedbyeventswhereonekaonpaircomesfromthedecayofaφ. Selecting
4
events with a kaon pair within 10 MeV of the φ mass, an enhancement in the invariant mass of the other kaon pair
.
1 close to threshold is observed and suggested to be due to the f (980) tail, but the low statistics and uncertainties
0
1 in the f (980) K+K− line shape prevent the authors to present a cross section for e+e− φf (980) using the
7 φK+K−0final s→tate. → 0
0
Inspired in the physics behind the radiative φ ππγ decay [6, 7, 8], a detailed theoretical study of e+e− φππ
:
v for pions in s-wave was performed in [9]. The pr→ocess e+e− φf has been also studied in the context of N→ambu-
0
i →
X Jona-Lasinio models [10]. In Ref. [9], it was shown that the tree level contributions through the ω φ mixing
are negligible and e+e− φππ proceeds through the production of off-shell KK¯ and K∗K pairs, the−successive
ar decay of the off-shell kaon→or K∗ into an on-shell φ and an off-shell K and subsequent KK¯ ππ scattering. The
starting point was the RχPT Lagrangian [11] supplemented with the anomalous term for th→e V′VP interactions.
The correspondingpredictions, valid for low virtualities of the exchangedphoton and low dipion invariantmass were
improved in two respects. First, the s-wave KK ππ amplitudes entering the loop calculations were replaced by
→
the full KK ππ isoscalar amplitudes as calculated in UχPT, which contain the scalar poles [12, 13] . Second,
→
the lowest order terms of the kaon form factor were replaced by the full kaon form factor as calculated in UχPT
[14] which describes satisfactorily the scarce data for energies around 2GeV [15, 16]. Likewise, the K∗K isoscalar
transition form factor arising from RχPT and the anomalous term, was replaced by the transition form factors as
extracted from data on e+e− K0K±π∓ [17].
→
Thef (980)couplesstronglytotheKK systemanditshouldcontributetothemechanismsstudiedin[9]inthecase
0
of φKK production. Therefore it is worthy to study this channel also. As discussed above, some experimental data
hasbeenreleasedfore+e− φK+K−. Wedevotethisworktothestudyofthisreactionintheframeworkdeveloped
in Ref. [9]. In contrast to→the φππ final state, the φK+K− final state is induced at tree level in this framework.
Furthermore, as noticed in [3], the f (980) pole is close to the threshold for the production of the dikaon system
0
and the loop contributions can be enhanced by this pole thus a complete analysis requires to calculate rescattering
contributions. Inthis concern,we knowthatthea (980)mesoncouples stronglyto the K+K− systembutnotto the
0
ππ system. Therefore, in addition to the f (980) contributions we also expect contributions from the a (980) meson
0 0
2
totheφK+K− finalstate. Thef (980)anda (980)poleslieslightlybelow2m ,hencetheircompleteshapesarenot
0 0 K
expected to be seen in this reactionbut their respective tails could give visible effects close to the reaction threshold.
Contributions from intermediate vector mesons, e+e− γ∗ φV φK+K−, are forbidden by charge conjugation.
→ → →
An important improvement with respect to the formalism used in [9] is the more accurate characterization of the
K∗K form factors. Indeed, recently, the cross section for e+e− K∗K was precisely measured in the 1.7 3 GeV
region[18]. WeusetheK∗K isoscalarandisovectortransitionfor→mfactorsasextractedfromthisdatainour−analysis
instead of the old data from [17].
The paper is organized as follows: In section II we outline the calculation. Results for the tree level contributions
to e+e− φK+K− are given in Section III. In section IV we adapt previous calculations for the rescattering effects
in the φπ→π final state to the φK+K− final state. Section V is devoted to the extraction of the K∗K transitions
form factors from data. In section VI we analyze the different contributions. Section VII is devoted to estimate the
intermediate X(2175) contribution. Our summary and conclusions are given in section VIII.
II. CALCULATION OF e+e− →φK+K−.
The calculation of the cross section for the e+e− φK+K− reaction is a non-trivial task. Indeed, the leading
→
electromagneticcontributions to this reactionare due to a single photon exchangein whose casewe needto calculate
the γ∗φK+K− vertex function for a hard virtual photon (√s & 2GeV). This is an energy scale far beyond the well
groundedcalculations basedon χPT , its (p4) saturatedversion(RχPT)or even the unitarized formalism,UχPT,
O
which is expected to be valid up to energies of the order of 1.2GeV; therefore it is not evident that one can perform
reliablecalculationsatthe energyofthe reaction. However,whatevertheresponsiblemechanismsforthereactionbe,
they must leave their fingerprint at low energies, i.e. for low photon virtualities in the γ∗φK+K− vertex function,
which can be calculated using the effective theory for QCD at low energies. We use this fact to attempt a reasonable
calculation of e+e− φK+K−. We calculate the γ∗φK+K− vertex function at low photon virtualities using RχPT
supplemented with t→he anomalous Lagrangian for the V′VP interactions. As a result we obtain the electromagnetic
part of the γ∗φK+K− vertex function dominated by form factors while the pure hadronic interactions are within
the scope of RχPT in the case of tree level contributions and involve the leading order on-shell χPT meson-meson
amplitudesinthecaseofoneloopcontributions. Theseresults,validatlowenergiesareimprovedusingtheunitarized
kaon form factors (which account for the scarce experimental data at the energy of the reaction) and experimental
data on the K∗K transition form factors. Similarly, the leading order on-shell meson-meson interactions are iterated
following [12, 13] to obtain the unitarized meson-meson amplitudes containing the scalar poles.
We start with the calculation of the tree level contributions to e+e− φK+K− within RχPT and consider also
intermediate vector meson exchange using the conventionalanomalous L→agrangianfor V′VP interactions. We follow
the conventions in [11] and the relevant interactions are given in Ref. [9]. The reaction e+e− φK+K− is induced
→
at tree level by the diagrams shown in Fig (1). In addition to the vertices quoted in Ref. [9] there is a tree level
γφK+K− point interaction whose vertex is given by
ΓγφK+K− = e√2 G FV k g e√2GV Q g , (1)
µαν − f2 V − 2 α µν − f2 α µν
(cid:18) (cid:19)
with the labels K+(p)K−(p′)γ(k,µ)φ(Q,α,ν) and all incoming particles.
III. TREE LEVEL CONTRIBUTIONS.
A straightforwardcalculation of the set of diagrams a) in Fig.(1) yields
e2√2Lµ F
i 1a= G Q G V g k ηαν (2)
− M − f2 k2 VTµν α− V − 2 µν α
(cid:20) (cid:18) (cid:19) (cid:21)
where k2 =(p++p−)2, Lµ v(p+)γµu(p−). The tensor is given by
µν
≡ T
(2p k) p′ (2p′ k) p
=g + − µ ν + − µ ν (3)
Tµν µν (cid:3)(k p) (cid:3)(k p′)
− −
where (cid:3)(p) p2 m2 +iε. It can be easily shown that this tensor satisfies
≡ − K
kµ Q ηαν =0 (4)
µν α
T
3
φ φ
K+
K− + K+ + φ
K+ K− K−
a)
φ φ
K− + K+
K+
K−
b)
φ φ
K− + K+
K+ K−
c)
FIG. 1: Tree level contributions to e+e− → φK+K− : a) pseudoscalar mesons exchange with point-like K+K−γ interaction
pluscontact term , b)pseudoscalar mesons exchangewith vectormeson mediated K+K−γ interaction, c) intermediate vector
mesons.
where ηαν denotes the antisymmetric tensor used to describe the φ field in RχPT. The second term in Eq. (2) is
explicitly gauge invariant, thus the γ∗φK+K− vertex function in the amplitude (2) is gauge invariant.
Diagrams b) in Fig. (1) involve the propagationof vector particles. These diagrams yield
i 1b= e2√2GV Lµ √2GVFVCVCVK+K− k2 W Q ηαν. (5)
− M − f2 k2  3f2 k2 M2 µν α
V=ρ,φ,ω − V
X
 
with the gauge invariant tensor
pδp′ p′δp
W = k2g k k ν + ν . (6)
µν µδ− µ δ (cid:3) (k p) (cid:3) (k p′)
(cid:18) V − V − (cid:19)
(cid:0) (cid:1)
Using the gauge invariance property in Eq. (4) it is possible to relate this tensor to as
µν
T
1
W = k2 k2g k k (7)
µν µν µν µ ν
2 T − −
(cid:2) (cid:0) (cid:1)(cid:3)
thus the amplitude can be rewritten as
e2√2G Lµ 1
i 1b= V F (k2) k2g k k Q ηαν (8)
− M − f2 k2 K+ Tµν − k2 µν − µ ν α
(cid:20) (cid:21)
(cid:0) (cid:1)
where e
F (k2)= 1 FV √2GVCVCVK+K− k2 = GVFV k2 + 1 k2 + 2 k2 . (9)
K+ 2 3 f2 k2 M2 2f2 m2 k2 3m2 k2 3m2 k2
V=ρ,φ,ω − V ρ− ω− φ− !
X
e
Adding up contributions from diagrams 1a) and 1b) we obtain
e2√2G Lµ 1
i = V FVMD(k2) F (k2) k2g k k Q ηαν (10)
− MP − f2 k2 K+ Tµν − K+ k2 µν − µ ν α
(cid:20) (cid:21)
+ e2√2 G FV Lµg k ηαν, e (cid:0) (cid:1) (11)
f2 V − 2 k2 µν α
(cid:18) (cid:19)
4
0.2
0.15
2
È
+K 0.1
F
È
0.05
0
1800 2000 2200 2400 2600 2800 3000
(cid:143)!s!!!
FIG. 2: Unitarized charged kaon form factor. Experimental points are taken from Ref. [15].
where
G F k2 1 k2 2 k2
FVMD(k2)=1+F (k2)=1+ V V + + . (12)
K+ K+ 2f2 m2 k2 3m2 k2 3m2 k2
ρ− ω− φ− !
e
Notice that the second term in Eq. (10) contains the vector meson contributions to the kaon form factor, but the
constant term due to the electric charge is missing. A similar result was obtained in [9]. This term should come
from Lagrangianswith higher derivatives of the fields, specifically from the term ∂αV ∂ fµν, which is absent in our
αν µ +
basic interactions . We will assume in the following that the constant term due to the charge is provided by such
missinginteractions;hence, wewillwrite FVMD insteadofF inthe secondtermofEq. (10). Also,notice thatthe
K+ K+
function FVMD(k2) accounts for the lowest order terms of the charged kaon form factor and describes it properly at
K+
lowk2. Thehighvirtualityofthe exchangedphotoninourperocessrequirestoworkoutthecomplete γK+K− vertex
function. The calculationof the kaon formfactors has been done in the context of UχPT in Ref. [14]. Although this
calculation misses the contributions of the first radially excited vector mesons which are important around 1.7 GeV,
theexistingexperimentaldataclosetotheφKK thresholdareproperlydescribedbytheunitarizedkaonformfactors
as showninFig (2) where the results for the unitarizedchargedkaonformfactor areplotted together with data from
the DM2 Coll. [15]. Thus in the following we will replace the lowest order terms so far obtained, FVMD(k2), by the
K+
fullunitarizedformfactorF (k2). Inadditiontothetermsassociatedtothekaonformfactorwegetacontactterm
K+
withthecombinationG FV . ThiscombinationissmallanditvanishesinthecontextofVectorMesonDominance
V − 2
[19]. Clearly,thistermcannotbeextrapolatedtohighphotonvirtualitieswithoutitsdressingbyaformfactor. The
impactofthis termonthe crosssectionfortheφππ finalstateuponits dressingwiththe kaonformfactorswasfound
to be very small in [9]. Its contribution turns out to be small also in this case when it is dressed with the charged
kaon form factor, thus we will skip it in the following. With these considerations the amplitude reads
e2√2G Lµ 1
i = V F (k2) k2g k k Q ηαν. (13)
− MP − f2 k2 K+ Tµν − k2 µν − µ ν α
(cid:20) (cid:21)
(cid:0) (cid:1)
In terms of the conventionalpolarization vector ην satisfying
i
Q ηαν(Q)=iM ην(Q), g k ηαν(Q)= (Q kg Q k )ην(Q), (14)
α φ µν α µν µ ν
M · −
φ
we finally obtain the tree level contribution from the exchange of pseudoscalar mesons as
ie2√2G M Lµ 1
i = V φ F (k2) k2g k k ην. (15)
− MP − f2 k2 K+ Tµν − k2 µν − µ ν
(cid:20) (cid:21)
(cid:0) (cid:1)
5
The amplitude for the diagrams c) in Fig. (1) is
i 1c = e2GT MK∗ Flo (k2)LµR ηαν. (16)
− M − √2 16 K∗+K− k2 µαν
(cid:18) (cid:19)
Here, we get the K∗+K− transition form factor to leading order as
Flo (k2)= GTCVK∗+K− FV CV 1 = FV G Mω + 3Mρ 2Mφ , (17)
K∗+K− V=ρ,ω.φ √2 3MK∗ k2−MV2 6 k2−Mω2 k2−Mρ2 − k2−Mφ2!
X
and R stands for the tensor
µαν
∆φηστ(k p) ∆φηστ(k p′)
R =g k ∆σβγδ(k)ǫ − + − ǫ , (18)
µαν µβ σ γδφη(cid:20) (cid:3)K∗(k−p) (cid:3)K∗(k−p′) (cid:21) σταν
with (cid:3)K∗(p)=p2−m2K∗, which is explicitly gauge invariant. It can be shown that
16i
R ηαν = ην (19)
µαν µν
MφMK∗V
where
(k p)η(k p)σ (k p′)η(k p′)σ
=kδǫ ǫφη +ǫ φ − − + − − Qα. (20)
Vµν µδφη(cid:20) αν σ αν(cid:18) (cid:3)K∗(k−p) (cid:3)K∗(k−p′) (cid:19)(cid:21)
TheamplitudeinEq. (16)containstheleadingordertermsfortheK∗+K− transitionformfactorvalidforlowphoton
virtualities. The photon exchanged in e+e− φK+K− has k2 & 2GeV and we should work out this form factor
→
for such high photon virtualities. In the numerical computations we replace the leading order terms by the complete
transition form factor, FK∗+K−(k2), extracted from recent data on e+e− → K0K±π∓, K+K−π0 to be discussed
below. With these considerations
ie2G Lµ
−iM1c =− √2 FK∗+K−(k2)k2Vµνην. (21)
IV. RESCATTERING EFFECTS
The final state φK+K− can get contributions from the K+K− K+K− rescattering. Furthermore, this final
state can also be reached through the production of φK0K0 and th→e rescattering K0K0 K+K−. Excitation of
highermassstatessuchasK∗K isalsopossibleandthefinalstateφK+K− canbeproduced→bytheinitialproduction
of an off-shell K∗0K0 (K∗+K−) pair, the subsequent decay of the off-shell K∗0 (K∗+) into a φK0(φK+) and the
rescattering K0K0 K+K− (K+K− K+K−). This section is devoted to the study of these contributions.
→ →
We calculate the rescattering effects to lowest order in the chiral expansion from RχPT supplemented with the
anomalous term for the V′VP interactions. These results are improved by the unitarization of meson-meson ampli-
tudesproposedin[12]whichdynamicallygeneratesthescalarresonances. TherelevantdiagramsinRχPT areshown
in Fig. (3), where for simplicity a shaded circle and a dark circle account for the diagrams i) plus j) and k) plus l)
respectively, which differentiate the direct photon coupling from the coupling through an intermediate vector meson.
The factorization of the meson-meson chiral amplitudes on-shell out of the loop integrals has been discussed in [9]
in the case of the φππ final state. The same considerations apply to the φK+K− final state and we refer the reader
to Ref. [9]for the details. To accountfor the whole rescatteringeffects in K+K− K+K− and K0K0 K+K− in
→ →
these diagrams we iterate the lowest order chiral amplitudes in the scalar channel, V and V to obtain the
K+K+ K0K+
unitarizedscalaramplitudes,t andt (inthe followingweskipthesuffix0 usedin[9]forthe scalarmeson-
K+K+ K0K+
meson amplitudes in order to handle isospin labels that will be necessary below and denote tI the MM M′M′
MM′ →
unitarizedamplitude in the isospinchannelI). Unlike the φππ case where only the isoscalarmeson-mesonunitarized
amplitudecontainingthef (980)poleisinvolved,inthecaseoftheφK+K−finalstatethenecessaryt andt
0 K+K+ K0K+
amplitudesarelinearcombinationsoftheisoscalarandisovectorscalaramplitudest0 andt1 . Thelastonecontains
KK KK
the a (980) pole. Notice that we are using the scalar amplitudes t and t instead of the full amplitudes
0 K+K+ K0K+
which in principle contain higher angular momentum contributions. This is expected to be a good approximation
for energies near the reaction threshold due to the small tri-momentum of the dikaon system. Furthermore, at these
6
l l k
−
l+Q
l+Q
l+Q l k
−
l k l
−
a) b) c)
l
l
l l k l+Q
−
d) e) f)
= +
j)
i)
=
+
k) l)
FIG. 3: Feynmandiagrams for e+e− →φK+K− in RχPT at one loop level.
energies the dikaon mass is near the f (980) and a (980) poles and the p-wave amplitudes (containing vector poles)
0 0
givevanishing contributionto this process due to chargeconjugation. Higherl (tensor) contributionsare expectedto
be important at much higher center of mass energy. The dominance of s-waves near threshold has been observed in
alike processes as J/ψ VPP [20].
→
The loop calculations go similarly to those of the φππ final state done in Ref. [9]. We adopt the same convention
of all external particles incoming and refer the reader to that work for the details. For kaons in the loops we obtain
ie2 Lµ
i = − A L(1)+B L(2) ην (22)
− MK 2π2m2 k2 P µν P µν
K
h i
with the Lorentz structures
L(1) Q kg Q k , L(2) =k2g k k , (23)
µν ≡ · µν − µ ν µν µν − µ ν
and
√2M G
A = φ V t F (k2)+t F (k2) I , (24)
P 2f2 K+K+ K+ K0K+ K0 P
√2M G (cid:2) (cid:3) m2
B = φ V t F (k2)+t F (k2) J + Kg . (25)
P 2f2 K+K+ K+ K0K+ K0 P 4k2 K
(cid:18) (cid:19)
(cid:2) (cid:3)
The loop functions are given by
1 x y(1 x)
I = dx dy − (26)
P Z0 Z0 1− mQ2K2 x(1−x)− 2mQ2K·k(1−x)y− mk22Ky(1−y)−iε
1 1 x y(1 2y)
J = dx dy − (27)
P 2Z0 Z0 1− mQ2K2 x(1−x)− 2mQ2K·k(1−x)y− mk22Ky(1−y)−iε
m2 σ+1
g = 1+log K +σlog , (28)
K − µ2 σ 1
−
7
στ αν
l
φη
w µ αβ γδ
l+Q
l k
−
FIG. 4: Feynman diagram for rescattering effects in e+e− →K∗K¯ →φKK¯ →φK+K−.
with σ = 1 4m2 /m2 . Notice that the only change from the final φπ+π− state [9], to the present case, is the
− K KK
substitution of the factor t F (k2)+t F (k2) by t F (k2)+t F (k2). It is convenient to
p K+π+ K+ K0π+ K0 K+K+ K+ K0K+ K0
write the meson-meson scalar amplitudes in isospin basis. With the conventions in Refs. [12, 14], we get
1 1
t = t0 +t1 , t = t0 t1 , (29)
K+K+ 2 KK KK K0K+ 2 KK − KK
(cid:0) (cid:1) (cid:0) (cid:1)
thus, in terms of the isoscalar (t0 , F0(k2)) and isovector (t1 , F1(k2)) meson-meson amplitudes and kaon form
KK K KK K
factors respectively we obtain
M G
A = φ V t0 F0(k2)+t1 F1(k2) I , (30)
P 2√2f2 KK K KK K P
M G (cid:2) (cid:3) m2
B = φ V t0 F0(k2)+t1 F1(k2) J + Kg , (31)
P 2√2f2 KK K KK K P 4k2 K
(cid:18) (cid:19)
(cid:2) (cid:3)
with the isoscalar and isovector form factors
F0(k2)=F (k2)+F (k2), F1(k2)=F (k2) F (k2). (32)
K K+ K0 K K+ − K0
Similarlyto the treelevelcontributions,the calculationofkaonloopsyields alsoa contacttermwiththe combination
G FV not shown in Eq.(22). For the same reasons we neglected the analogous term in the previous section, we
V − 2
will also neglect it here.
The process e+e− φ K+ K− can also proceed through the diagrams shown in Fig. (4). Again, the calculations
→
are similar to the case of φππ in the final state calculated in Ref. [9] and we refer to this work for details on the
notation and conventions. The amplitude from the diagram in Fig. (4) gets contributions from charged and neutral
K∗K in the loops. We obtain the amplitude from these diagrams as
ie2 Lµ
i = − A L(1)+B L(2) ην (33)
− MV 2π2m2 k2 V µν V µν
K
h i
with
G
AV = 4√2 FK∗+K−(k2)tK+K+ +FK∗0K0(k2)tK0K+ IV, (34)
GM(cid:2)2 (cid:3)
BV =− 4√2φ FK∗+K−(k2)tK+K+ +FK∗0K0(k2)tK0K+ JV (35)
(cid:2) 1 m2 (cid:3)
I =m2 I I +2+ log K +Q k J . (36)
V K G− 2 2 µ2 · V
(cid:18) (cid:19)
HereFK∗+K−(k2)standsfortheleadingordertermsintheK∗+K− transitionformfactorinEq. (17)andFK∗0K0(k2)
denotes the K∗0K0 leading order terms of the neutral transition form factor given by
F G M 3M 2M
F (k2)= V ω ρ φ . (37)
K∗0K0 6 k2−Mω2 − k2−Mρ2 − k2−Mφ2!
8
The loop functions are given by
1 x y(1 x)
J = dx dy − (38)
V Z0 Z0 1 Q2 x(1 x) 2Q·k(1 x)y k2 y(1 y) (m2K∗−m2K)(y x) iε
− m2K − − m2K − − m2K − − m2K − −
I = 1dx xdylog[1 Q2 x(1 x) 2Q·k(1 x)y k2 y(1 y) m2K∗ −m2K (y x) iε] (39)
2 − m2 − − m2 − − m2 − − m2 − −
Z0 Z0 K K K (cid:0) K (cid:1)
m2 σ+1
I = 2+log K +σlog . (40)
G − µ2 σ 1
−
Notice again that the only change from the final φπ+π− state to the present case is the replacement of the factor
tK+π+FK∗+K−(k2)+tK0π+FK∗0K0(k2) by tK+K+FK∗+K−(k2) +tK0K+FK∗0K0(k2). In terms of the isoscalar and
isovector amplitudes and transition form factors we get
G GM2
AV = 8√2 t0KKFK0∗K(k2)+t1KKFK1∗K(k2) IV, BV =− 8√2φ t0KKFK0∗K(k2)+t1KKFK1∗K(k2) JV. (41)
(cid:2) (cid:3) (cid:2) (cid:3)
with
FK0∗K(k2)=FK∗+K−(k2)+FK∗0K0(k2), FK1∗K(k2)=FK∗+K−(k2)−FK∗0K0(k2). (42)
Finally, taking into account both pseudoscalars (Eq.(22)) and vectors (Eq.(33)) in the loops we obtain the total
amplitude as
ie2 Lµ
i = − A L(1)+BL(2) ην (43)
− M 2π2m2 k2 µν µν
K
h i
where
A=A +A , B =B +B (44)
P V P V
with the specific functions in Eqs.(30,31,41). Recall these results are valid for ingoing particles. For the numerical
computations in the following section we reverse the momenta of the final particles to obtain
ie2 Lµ
i = I L(1) J L(2) ην (45)
− ML 2π2m2 k2 µν − µν
K
h i
with
I =A A , J =B B (46)
P V P V
− −
where the functions AP, AV, BP, BV are obtaeined ferom the untileded fuenctions in Eqs. (30,31,41) just changing the
sign of Q k in the integrals I , J , J , and I in Eqs. (26,27,38,39).
P P V 2
·
e e e e
V. K∗K TRANSITION FORM FACTORS
Our results involve the K∗K isoscalar and isovector transition form factors at the center of mass energy of the
reaction. Our calculation based on lagrangians suitable for low energies yields the lowest order terms in the chiral
expansion for these form factors. Since a calculation of these form factors at such huge energies as √s 2 GeV is
≥
beyondthe scope ofthe effective theoriesused here,we consider ourcalculationas a convenientprocedureto identify
the main physical effects ( form factors and meson-meson amplitudes) and must find a way to enlarge the range of
validity of the calculation to high energies. We do this by replacing the leading order terms in the transition form
factors by an appropriate characterizationof these form factors at the energy of the reaction. In the case of the φππ
finalstate,onlytheisoscalartransitionformfactorisrequiredandin[9]itwasextractedfromdataone+e− KKπ,
→
at √s= 1400 2180 MeV reported in [17]. In the present case, also the isovector transition form factor is required.
−
This form factor cannot be extracted from this data due to the low statistics. Indeed, in [17] it is assumed that the
isovector amplitude is dominated by a ρ′ with a mass and width fixed to mρ′ = 1570 MeV and Γρ′ = 510 MeV in
the interpretation of the data. Fortunately, high precision measurements for the cross section of e+e− K+K−π0,
→
9
K0K±π∓ were recently released where the contributions from intermediate K∗are identified [18]. The first reaction
involves the charged K∗K transition form factor while the second gets contributions of both charged and neutral
transitionformfactors. Usingthis data,the isoscalarandisovectorcomponentsofthe crosssectionfore+e− K∗K
→
at √s = 1400 3000 MeV were extracted along with an energy dependent phase which signals another resonance
−
below 2 GeV which could not be resolved [18].
We extract the form factors from this data as follows. First we calculate the γ(k,µ)K∗(q,ν)K(p) vertex function
in RχPT and replace the lowest order terms in the transition form factor by the full one to obtain
Γµν(k,q)=ieFK∗K(k2)ǫµναβkαqβ (47)
withallincomingparticles. Astraightforwardcalculationofthee+e− K∗K crosssectionusingthiseffectivevertex
→
yields
πα2
σ(s)= 6s3|FK∗K(s)|2λ32(s,m2K∗,m2K) (48)
where
λ(m2,m2,m2)=(m2 (m m )2)(m2 (m +m )2). (49)
1 2 3 1− 2− 3 1− 2 3
In Ref. [18] a fit was done to the isovector and isoscalar components of e+e− K+K−π0, K0K±π∓using also data
on the cross section for the production of φη where the φ′ peak is visible. T→he analysis below 2 GeV required the
introductionofanenergydependentrelativephasepointingtothe existenceofanunresolvedresonanceinthisenergy
region,inadditiontotheρ′ andtheφ′. Here,weareinterestedonlyinanappropriatecharacterizationoftheisoscalar
and isovector form factors in the 2 3 GeV region where the effect of this phase should be small. The experimental
−
points for the squared absolute value of the form factors are obtained using Eq. (48) and tables VI and VII of Ref.
[18]. Physically, these form factors are dominated by the exchange of resonances; hence in their characterization we
complement the lowest order terms obtained in RχPT with the exchange of heavy mesons
FK0∗K(k2)= FV3G k2M−ωMω2 − k22−MMφ φ2!+b0 k2−m2φ−′ 2+mi√2φ′sΓφ′(s)!, (50)
FK1∗K(k2)= FV3G(cid:18)k23−mmρ 2ρ(cid:19)+b1 k2−m2ρ′3+mi2ρ√′ sΓρ′(s)!, (51)
with the energy dependent widths used in [18]
Γφ′(s)=Γφ′"PPKK∗K∗K(m(s2φ)′)BKφ′K∗ + PPφηφ(ηm(s2φ)′)Bφφη′ +(1−BKφ′K∗ −Bφφη′)#
Γρ′(s)=Γρ′"PP4π4(πm(s2ρ)′)B4ρπ′ +(1−B4ρπ′)#,
where
λ(s,m2,m2) 3/2 s 16m2 3/2
(s)= V P , (s)= − π . (52)
VP 4π
P s P s
(cid:18) (cid:19) (cid:0) (cid:1)
The values forthe constantsappearinghereareextractedfromthe centralvalues ofTabelXV in[18]as Bφ′ =0.5,
KK∗
Bφφη′ =0.20, B4ρπ′ =0.65, mφ′ =1723 MeV, Γφ′ =371 MeV, mρ′ =1504 MeV, Γρ′ =438 MeV. The parameters b0,
b arefittedtothedatafortheisoscalarandisovectorcrosssectionsreportedintablesVIandVIIof[18]respectively.
1
The fit to the isoscalar cross section yields b = 0.2487 10−3MeV−1. The fit to the isovector cross section yields
0
b = 0.2551 10−3 MeV−1. Notice that thes−e values×are nearly equal, pointing to the φ′ and ρ′ as members of
1
− ×
an SU(3) nonet. Our results for the corresponding cross sections are shown in Figs. (5,6), where the shadowed
bands correspond to the 1σ regions for the paramenters b and b . A good description of the isoscalar cross section
0 1
is obtained but more accurate data on the isovector e+e− K∗K cross section is desirable. The discrepancy in this
case is due to the effects another intermediate resonance a→round √s=2GeV in e+e− K∗K which however could
→
not be resolved in [18].
As a cross check, we calculate the full cross section for e+e− K+K−π0 from the exchange of K∗ which involves
→
only the charged transition form factor and obtain a proper description of data in Table III of Ref. [18] up to
√s=3GeV.
10
10
8
6
4
2
0
1.5 2.0 2.5 3.0
FIG.5: Crosssectionfore+e− →K∗K intheisoscalarchannel. Thebandcorrespondstothe1σ regionofourparametrization
of the form factors in Eq.(50). Data points are taken from Table VI of Ref. [18].
VI. NUMERICAL RESULTS AND DISCUSSION
With respect to the φππ production with s-wave pions studied in [9], there is now the novelty of the tree level
contributions in φK+K− production with K+K− pairs in all possible angular momentum states. In the case of
loops, our calculations consider only s-waveK+K− in the final state because we are using the unitarized amplitudes
for KK K+K− which per construction are projected onto well defined angular momentum (l = 0) and isospin
→
(I =0,1). In the Appendix we give details on the integration of phase space and state our conventions. The dikaon
spectrum turns out to be
dσ 1 m E+ π 2π
= KK dE dcosθ dϕ 2, (53)
dmKK (2π)4 16s23 ZE− Z0 V Z0 |M|
where
= + 1c+ (54)
P L
M M M M
andthese amplitudesaregiveninEqs. (15,21,45)respectively. We referthe readertothe appendixforfurtherdetails
onthenotationandintegrationofphasespace. Weevaluatenumericallytheintegralsandthedifferentialcrosssection
using the physical masses and coupling constants. We remark that all the parameters entering this calculation have
been fixed in advance and, in this sense, there are no free parameters. In our numerical analysis we use m = 495,
K
m = 1019.4, α = 1/137, G = 53MeV, F = 154MeV, f = 93 MeV, and G = 0.016MeV−1. Concerning the
φ V V π
rescattering effects we remark that the K+K− K+K− and K0K0 K+K− scattering between the kaons in
→ →
the loops and the final kaons takes place at the dikaon invariant mass independently of the value of √s and of the
momenta in the loops. As a consequence, when replacing the lowest order terms for this amplitude by the unitarized
amplitude, we can safely use the results of [12, 13] and take a renormalization scale µ = 1.2 GeV for the function
G [13], in spite of the fact that the reaction e+e− φK+K− takes place at a much higher energy √s 2GeV.
K
→ ≥
The unitarized amplitudes naturally contain the scalar poles and there is no need to include explicitly these degrees
of freedom in the calculation. We obtain the dikaon spectrum shown in Fig. (7) for √s = 2010 3000 MeV. In
−