Table Of ContentTransition form-factor of γ 3π in nonlocal quark model.
→
A. S. Zhevlakov∗
Department of Physics, Tomsk State University, 634050 Tomsk, Russia
(Dated: April4, 2017)
Thetransition form factorofagammaintothreepionsF3π isstudiedinaframework ofnonlocal
chiral quark model. In local limit the result is in agreement with chiral perturbative theory and
reproduce Wess-Zumino-Witten anomaly. Nonlocality does not change a value of transition form
factor in chiral limit. On the physical threshold of reaction the form factor F3π obtained is in a
good agreement with experimentaldata if thecurrent quark mass is taken intoaccount in model.
I. INTRODUCTION Another processes which are also given entirely in
7
terms of the electric charge e and the pion decay con-
1
0 Starting from a paper of J. Steinberger in 1949 [1] stantfπ isthereactionofγπ± π±π0 ore+e− γ∗
2 anomalies have played an important role in investiga- π0π−π+. These processes has→also a connection→to th→e
r tionofstrong-interactionphysicsatlow-energies. Incase WZW anomalous effective actionand amplitude of reac-
p of chiral model, anomalous Ward identities are due to tion have the same following form:
A
Wess-Zumino-Witten (WZW) effective action [2]. WZW
3 actioncan be constructed in terms of pseudoscalarfields A(γπ− →π−π0)=−iF3π(s,t,u)ǫµναβǫµpν0pα1pβ2, (3)
- Goldstone bosons and interaction vertices involving an where ǫµ is polarization of incident photon and pi are
] odd number of Goldstone bosons. Using a topology ar-
h momenta of pions. In chirallimit in low-orderby quark-
guments, E. Witten showed [3] that this action can be
p loops, form-factor of this amplitude is independent from
quantized and i.e. a multiple of an integer parameter
- Mandelstam variables s,t,u and have a simple form [12]
p n which associated also with baryon number in soliton
he moJd.eWlse.ssandB.Zuminoconsideredthehadronicsystem F3π(0,0,0)= 4πe2f3 =9.72GeV−3. (4)
[ π
in presence of external vector and axial vector fields[2].
3 As a result the anomalous Ward identities and vertex of The processthatdescribedthis amplitude wasmeasured
v interactionofGoldstonebosonswithexternalvectorfield for the first time at the IHEP accelerator (Serpukhov)
1 at the 40-GeV negative-pionbeam [13]. The experiment
were obtained. All amplitudes that followfrom ofWZW
7 wasbasedonpionpairproductionbypionsinthenuclear
action are givenentirely in terms of the electric charge e
1
Coulomb field via the Primakoffreaction
7 and the weak pion decay constant fπ.
0 A well-studied process that follows from WZW action π− + (Z,A) π−′+ (Z,A) + π0. (5)
. isdecayofaneutralpseudoscalarmesonintotwophotons →
1
0 or production of a neutral pseudoscalar (PS) mesons in The value of F3π extracted from the experiment is
7 e+e− collisions. This transition form-factor was studied
1 using a number of different approaches: chiral perturba- Fexp =(12.9 0.9 0.5)GeV−3. (6)
3π ± ±
: tion theory (ChPT), vector meson dominance approach
v
This expression still has a small difference from the the-
[4, 5], sum rules [6, 7], different local andnonlocal quark
i
X models [8, 9]. According to WZW action, the amplitude oretical value given by low-energy theorem(4). The one-
photonexchangeisadominatingprocessinthisreaction.
r of this reaction has the following form
a Onecanexpectthatthehigherprecisioninmeasuringof
A(π0 γγ)=F (M2 )ǫµναβǫµkνǫαkβ, (1) this value will be obtained from the future experiments.
→ γγ π0 1 2
Therefore, there is a need in more accurate theoretical
whereǫi andki -polarizationsandmomentaofphotons,
j j estimations of this value.
transition pion-photons form factor
From the experiment side of reaction (5) will be mea-
e
suredinCERNCOMPASSexperiment[14]forscattering
F (0)= . (2)
γγ 4π2fπ kaons and pions on nuclear target. There exist theoreti-
cal predictions on the value of the formfactor (5) in the
The weak pion decay constant is f = f [1+O(m )] =
π 0 q
framework of different approaches, e.g., low-energy the-
92.4MeV. ThequarkmasscorrectioninChPTisasmall
[10] but plays an important role in Dalitz decay π0 orem, ChPT (with and without taking to account the
γe+e− As it was shown in reference [11]. On a mas→s- q2-dependence and electromagnetic corrections)[15], dis-
persion relations and others.
shell form-factor F have a corrections due to mass of
γγ
The present work will be focus of the transition form-
pion.
factor for γ∗ π+π0π− which will be described in the
→
framework of a nonlocal chiral quark model (NχQM).
This model is a nonlocal extension of the Nambu-Jona
∗ [email protected] Lasinio (NJL) model [16–20]. The NχQM model has
2
tested in different regions of particle physics: meson dy- andtrickofGaussianintegral. Integratingoutthequark
namics [8, 19], deconfinement [21], dense matter at high fields one can see that produced functional have a form:
temperature and/or intense magnetic fields [22, 23].
Inthiswork,the useofthis modelresultsinreproduc- Z = D~πDσ exp[ S(σ,π)], (10)
tionoflow-energytheoremfortheprocessphotonintoto Z − E
three pions. Due to nonlocality of this model the result
where bosonisated action is
obtained is in a good agreement with the experimental
data from Serpuhov [13].
1 d4p
This paperis organizedasfollows. Insect. II the non- S(σ,π) = lndet(D)+ (σ2+~π2). (11)
E − 2GZ (2π)4
local quark model is briefly considered. In sect. III the
calculation of transition form-factor of photon into tree The operator D in momentum space can be written as
pionsdecayareperformedintheframeworkofthismodel
andsomepropertiesofresultsobtainedarediscussed. In D =( pˆ m )(2π)4δ(p p′)+f(p2)f(p′2)(σ+τaπa),
c
sect. IV the summary is given and some conclusions are − − −
made. Technical details can be found in appendix. where f(p) is Fourier transform of the form factor f(x).
Assuming that scalar field σ has a non-trivial vacuum
average, while the mean field (MF) values of the pseu-
doscalarfieldsπaarezero,itispossiblerewritteninterms
II. NχQM
of new scalar and pseudoscalar fields as follows:
The Lagrangianof the SU(2) SU(2) nonlocal chiral σ(x)=m +σ(x); πa(x)=πa(x). (12)
× d
quark model has the form [16, 20]
Thisproposalofnon-trivialvacuumaverageofscalarfield
NχQM=q¯(x)(i∂ˆ mc)q(x)+ (7) spontaneouslybreaks of symmetry. Parametermd is dy-
L −
G namical mass of quark.
2[JSa(x)JSa(x)+JPa(x)JPa(x)]. Bosonized effective action can be expanded in powers
of the meson fluctuations and the expression takes the
whereq(x)arethequarkfields,m isthediagonalmatrix form
c
of the quark currentmasses [24] and G is the four-quark
coupling constant. S(σ,π) =SMF +Squad+... (13)
E E E
The nonlocal structure of the model is introduced via
the nonlocal quark currents For calculation of considered transition form-factor one
needstoconcentrateonmeanfieldactionperunitvolume
and action in the quadratic term for the pseudoscalar
Ja (x)= d4x d4x f(x )f(x )q¯(x x )Γa q(x+x ),
S,P Z 1 2 1 2 − 1 S,P 2 fields. These are given by
ΓaS =τa,ΓP =iγ5τa, (8) SMF d4p m2
E = 4N ln[p2+m2(p2)]+ d, (14)
V(4) − cZ (2π)4 2G
where f(x) is a form factor reflecting the nonlocal prop-
ertiesoftheQCDvacuumandτa arePaulimatrices. For Squad = 1 d4p G−(p2)~π(p) ~π( p).
simplicity in this work we do not consider an extended E 2Z (2π)4 · −
model [19, 25] that includes other structures besides the
pseudoscalar (P) and scalar (S) ones. New notation are introduced here
The form factor f(x) characterizing the composite
m(p2)=m +m f2(p2), (15)
structure of hadron is an unknown function. The choice c d
1
of this form factor can be sensitive for physical observ- G−(p2)= 8N Π (p2), (16)
c a
ables [26]. To simplify the calculations the Gaussian G −
[20,26]formoftheformfactorfunctionwillbeused. This
where the mass of quark has in dependence on momen-
function describe a nonlocality of interaction of quark tum. The polarization operator Π (p2) is
a
field with mesons.
f(x)=Z (2dπ4k)4eixkf(k2), f(k2)=ek2/Λ2, (9) Πa(p2)=Z (d24Eπk)4fk2+kf+2k2−+(cid:2)m(k2+(k·+2k−))±k−2m+(km+2)2m(k(−2k)−2)(cid:3),
(cid:2) (cid:3)(cid:2) (cid:3)(17)
where Λ is a parameter cutoff. Note that this form is with k± =k±p/2 and fki =f(ki2), index a corresponds
not the only one that can be used for parametrization to different channels of meson. The upper sign in nu-
quark-antiquark potential [19, 26]. meratorcorrespondstothepseudoscalarchannelandthe
Observable degrees of freedom are meson states. Pro- lower sign corresponds to the scalar channel. The inte-
cedure of introducing of meson states is due to using gration here and later is gone on Euclid space d4k and
E
properties of partition function Z = DqDq¯exp[ S ] k2 = k2.
− E − E
R
3
A. Fitting the model
0,32
After introducing the dynamical mass of quarks, the
model have four parameters: current m and dynamical
c 0,30
m quarkmasses,four-quarkinteractionconstantG and
d
cutoff parameter Λ. These parameters are not indepen- V]0,28
dent. There have connections which can be obtained as Ge
follows. Using the properties of action S for composite 3 , [0,26
operators,one canconstruct anequation for the size pa- 1/
>
rameter (so-called gap equation): qq0,24
<
-
0,22
∂S
=0. (18)
(cid:28)∂σ(cid:29) 0,20
0
0,18
The latter relation leads to the following expression for 0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,32 0,34 0,36 0,38
dynamic quark mass:
md, [GeV]
m =8N G d4Ep f2(p2)m(p2). (19) FIG.1. Behaviorofquarkcondensate−hq¯qi1/3 from dynam-
d c Z (2π)4 p2+m2(p2) ical mass of quark in NχQM model. Black solid line corre-
spondsachirallimit. Reddot-dashedlineshowsadependence
One parameter can remain free, for example, dynamical at physical current quark mass. Dashed box corresponds the
emperical boundsfrom [29, 30].
mass. This parameter can vary in its appropriate phys-
ical range. Two parameters: current m and G or Λcan
c
befittedusingpionmassandwidthofpionintotwopho-
tons decay. Last parameter can be found from the gap B. Nonlocal vertices
equation.
The chiral condensates are given by the vacuum ex- After we apply of mechanism of spontaneous sym-
pectation values q¯q = u¯u = d¯d . By performing the metry breaking, the nonlocal interaction of quarks
h i h i h i
variationofmeanfieldpartitionfunctionwith respectto with meson fields generates corresponding quark self-
the correspondingcurrent quark masses is obtained that interaction vertices and interaction vertices with gauge
fields. These vertices can be incorporated in the model
d4p m(p2) m usingSchwingerphasefactor. The theoryofnonlocalin-
hq¯qi=−4NcZ (2Eπ)4 (cid:18) D(p) − p2+cm2(cid:19), (20) teraction gauge fields with quark and meson fields was
c constructed in [34] and later was used to study the non-
local quark model [8, 16, 18, 25, 35].
where D(p)=p2+m2(p2).
The nonlocal vertex of interaction quark-antiquark
Value of chiral condensate of light quarks is one of with external field can be written as:
the fundamental parameters of non-perturbative QCD
and chiral symmetry. Detailed analysis of quark con- Γ (q)=γ (p +p ) m(p ,p ), (21)
µ µ 2 1 µ 1 2
densate behavior in nonlocal quark model with different −
parametrizations was provided in [27]. Typical interval wherep andp =p +qaremomentaofquarks,qismo-
1 2 1
forphysicaldynamicalmassofquarkintheNχQMmodel mentum of external field [8, 16, 35]. Note that one can
is takenin range 200 350MeV as reproducesempirical generate infinite number interaction vertices of quarks
bands for light quark÷condensate q¯q 1/3 200 260 with external gauge fields. For interaction of quark-
−h i ≃ ÷
MeV[28–30]. InadditionphysicalrangeinNχQMmodel antiquark with scalar or pseudoscalar mesons, vertices
with Gaussian function of description a nonlocality of take forms:
quark interaction with meson has qualitative agreement
withtheresultsofquarkcondensatefromlatticecalcula- Γa =g (q2)τaf(p2)f(p2), (22)
σ σ 1 2
tion [31, 32] and result from renormalizationgroup opti- Γa =g (q2)iγ τaf(p2)f(p2), (23)
π π 5 1 2
mizedperturbationmethod[33]. Behavioroflightquark
condensate from dynamical quark mass md in NχQM where p1 and p2 = p1 + q are momenta of quarks, q
model is shown in Fig.1. Behavior of quark condensate is momentum of a meson, g (q2) and g (q2) are con-
σ π
atphysicalvalue ofcurrentquarkmasshassameviewof stants which describe a renormalization of scalar or
curve as one in chiral limit but lies just below. pseudoscalar meson fields accordingly. The constants
The error bar of estimation different values in frame- g (q2) are obtained by solving Bethe-Salpeter equa-
(σ,π)
work of NχQM model comes from the band in region of tion (BSE) [8, 16, 18] in the ladder approximation. As a
physical dynamical mass of quark. resultofsolvingof BSE for a propagatorofa meson, the
4
corresponding quark-pion constant can be expressed as III. FORM-FACTOR OF F3π
In this section, the transition form-factor photon into
1 g(2σ,π)(p2) three pions in the framework of NχQM is be described.
= , (24)
G+Π (p2) p2 m2 This form-factor follows from anomalous Ward identity.
− (σ,π) − (σ,π)
Herewereproducethe dependenceofthisform-factoron
quarkmass in NχQMmodel. Note that this dependence
where m is mass of sigma or pionmesons. Onmass- is absent in local limit.
shell pion(σ,,πq)uark-pion constant can be fixed [17, 36, 37] In general, the amplitude of γ∗ π+π0π− processes
→
can be written as a sum of six diagrams.One of these
diagrams is shown on Fig.2 and the others can be ob-
1 ∂Π (p2) tained from it by permutations of pion legs. There are
(σ,π)
= , (25) three groups of diagrams which have a symmetry under
g(2σ,π)(m2(σ,π)) ∂p2 (cid:12)(cid:12)(cid:12)p2=m2 exchange of direction of quark momentum in quark loop
(cid:12) (σ,π) k k.
(cid:12) →−
Inframeworkofnonlocalquarkmodelwithlinearcase
where Π (p2) is polarization operator that is defined ofbosonization,contactednonlocalverticesofinteraction
(σ,π)
above. Constant g (q2) has a simple connection with pion with quark-quark-photon exist [9, 18, 38]. These
π
verticesdon’tmakeacontributioninF transitionform
weak pion decay constant through Goldgerger-Treinman 3π
factor because terms which contain them vanish after
(GT) relation in chiral limit of model.
Dirac matrices trace calculation.
The amplitude of transition of gamma in three pions
γ π can be written as
A(γ π+π0π−)= iF (s,t,u)ǫµναβǫµpνpαpβ, (26)
→ − 3π 0 1 2
where p are momenta of pions, ǫµ is a polarization of
i
photonandF (s,t,u)isaLorentzscalarfunctionofthe
3π
Mandelstamvariables. Itcanbedefinedfromthreetypes
of diagrams in different kinematics:
π π F3π(s,t,u)=F1(s,t,u)+F2(t,s,u)+F3(u,t,s),(27)
FIG. 2. Feynman diagram which describing transition form- where s,t,u - are Mandelstam invariance variables. The
factor γ∗π− →π0π−. All vertices are nonlocal. first function of transition form-factor can be calculated
from the Feynman diagramsFig.2 and in NχQM model
have a form
F (s,t,u)=eN d4Ek gπ(p20)gπ(p21)gπ(p22)fkfk2+p1fk2−p0fk−p0−p2Trf[Q(τ−τ3τ++τ+τ3τ−)]
1 cZ (2π)4 D(k)D(k+p )D(k p )D(k p p ) ×
1 0 0 2
− − −
4 m(k2)[A+1 B] m((k p )2)[C +A]+m((k+p )2)C+m((k p p )2)B , (28)
0 1 0 2
× − − − − −
(cid:8) (cid:9)
where D(k)=k2+m2(k2), Q is a chargematrix ofquark and τ± =(τ1 τ2)/√2 are combinations ofPaulimatrices
±
that follow from pion fields. A coefficients A,B,C can be found in Appendix A. Structure functions F (s,u,t) and
2
F (t,s,u) have a connection with F (s,t,u) by crossing symmetry, namely:
3 1
F (t,s,u)=F (s,t,u)(p ⇆ p ,τ3 τ+) and F (u,t,s)=F (s,t,u)(p p ,τ3 τ−). (29)
2 1 0 1 3 1 0 2
− ↔ ↔− ↔
In low energy limit, when kinematic invariants s = t = u = 0 and performing the chiral expansion over m the
c
value of transition form factor has a form
d4k (m (k2) m′(k2)k2)
F (0,0,0)=eN N g3 E f6(k) 4 0 − 0
3π c f πZ (2π)4 (cid:26) (cid:20) D(k)4 (cid:21)
(m2(k2) m (k2)m′(k2)k2) 1 1
32m 0 − 0 0 . (30)
− c(cid:20) D(k)5 − 8D(k)4(cid:21)(cid:27)
Here m(k2) m (k2) = m f2(k2), g = g (0) and second term in eq.(30) has a dependence from current mass of
0 d π π
→
5
quark. In chiral limit current quark mass m is zero this form factor takes a form
c
eN N d4k 4m4(k2) 4m′(k2)m3(k2)k2
F (0,0,0)= c f E 0 − 0 0 , (31)
3π f3 Z (2π)4 (cid:20) D(k)4 (cid:21)
π
∂m (k2)
where m′(k2) = 0 . Goldgerger-Treinman rela-
0 ∂k2
tionatthequarklevelwhichholdsinchirallimitofquark
15
model, f =g /m , is used [20, 39, 40].
π π d
When the parameter of nonlocality tends to infinity 14
Λ , f(k2) 1 and m′(k) = 0, m(k2) = m in
d
→ ∞ →
chiral case. Then the integral in Eq.(31) can be solved 13
]
analytically: -3 V
e 12
∞ k2m4 1 G[
dk2 d = . (32) ,
Z (k2+m2)4 6 F311
0 d
In this particular limit, the eq.(31) reproduces WZW 10
form-factor [2]:
9
eN N e
F = c f = 9.72(0.09)GeV−3. (33)
3π 24π2f3 4π2f3 ≃ 8
π π 0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,32 0,34 0,36 0,38
Behavior of the form factor F in NχQM model is md, [GeV]
3π
different on pion mass-shell and in chiral limit due to
nonzero current quark mass. The dependence of F FIG. 3. Dependence of transition form-factor γ∗π− →π0π−
3π
form-factoron dynamical quark mass for both cases of on dynamical mass of quark md. Solid black line show a be-
pionmass-shelland chirallimit is shownonFig.3. Aver- haviorofF3π inchirallimit,dash-dotredlinecorrespondsto
thetransitionform-factoronthephysicalthresholdofprocess.
aged quantity of F at low energy in chiral limit is
3π
The area between the olive dot lines show a range from Ser-
F (0,0,0)=9.789(0.4)GeV−3. (34) pukhovexperiment[13]. Bluedashedlinecorrespondsalimit
3π
valueF3π whenquark-mesonnonlocalityformfactortendsto
Therefore, we can conclude that the presence of non- zero, f(p2) →0, in NχQM model or quantity obtained from
locality does not change the picture in chiral limit and low-energy theorem.
the obtained value of F is in agreement with the one
3π
obtained from the low-energy theorem.
For physical masses of pions, transition form fac- pionatnonzerocurrentmassofquarkplaysanimportant
tor should be calculated on the physical threshold for role here.
q2 = 0 and s + t + u = 3m2. In this case, kinemat- As in the case of pion transition form factor into two
π
ics variables [12] take the form sthr = (mπ− +mπ0)2, photons [10], F3π form factor is slowly changed with the
tmptehπrr−tu=mrbπ−a0mt−ioπn−mmb2πy02π)0/m/((m2mπtπr−−an++simmtioππ0n0)).foaTrnmhduufsatcihntrol=rowFmetsπht−ro(hmrads2πe−rtho−ef gmInraoswlostwohefsoqtfudoayrrdnkearimsoiacfamdleomcroaemsismpoopfsoqitruitoaannrkti.neIffnpeccoltwusfeoirorsntoohffeecmluerocrtderenolt-.
π 3π
form: magnetic constant,nonlocal interactionexternalelectro-
magnetic fields with quarks does not play a significant
d4k
Fthr(sthr,tthr,uthr)=eN N g3(m2) E f6(k2) roleandthe importantcontributionis dueto localvertex
3π c f π π Z (2π)4 × of interaction of quarks with a photon. Same effect oc-
4m(k2) 4m′(k2)k2 curs for transition form factor of pion into two photons
×(cid:20) D−(k)4 (cid:21)+O(m2π). (35) [8]. Nonlocalityofmeson–quarksverticesprovidedbyex-
istence of form factors f(p2). The model in chiral limit
Notethatthisformissimilartotheoneobtainedinchiral reproduce the form of expression F that follows from
3π
limit. The corrections due to pion mass are suppressed. theWZWaction. ThevalueF inNχQMinchirallimit
3π
Dependence on current quark mass leads to a following is in agreement with the one from low energy theorem.
change of transition form factor:
Fthr(sthr,tthr,uthr)=11.48(0.58)GeV−3. (36)
3π IV. OUTLOOK
Directviolationofchiralsymmetrybyincluding nonzero
currentquarkmassleadstoincreaseofvalueoftransition In article the transition form-factor of γ∗ π+π0π−
→
formfactorF (seeFig.3). Theshiftofdecayconstantof which reproduce a WZW term from anomalous Ward
3π
6
identitiesinchirallimitofmodelwasstudied. Itisshown for close reading. I thankful to M. V. Polyakov for mo-
that the nonlocality does not play a significant role for tivation to study this subject. The work is supportedby
theformfactorF inchirallimit. onthephysicalthresh- Russian Science Foundation grant (RSCF 15-12-10009).
3π
old value transition form factor in nonlocal quark model
Fthr = 11.48(0.58)GeV−3 has a good agreement with
3π
experimentalwhichwasmeasuredbyAntipovetal. [13].
The dynamical quark mass has in the typical interval Appendix A: Kinematic functions
200 350 MeV for physical mass, and the current mass
÷
of quark has a typical values in range of 4 9 MeV.
Functions A, B, C that appeared in eq.(28) after
÷
The resultobtainedfor F has anagreementwith re-
3π rewriting dependence of Lorenz-index loop integral in
sults from dispersion analysis in two-loop [41], in frame-
terms of external fields have the forms:
workofintegralequationapproach[42]andfromthecase
of dispersive representations with the ππ P-wave phase (kp )(p2p2 (p p )2)
shift [43]. ChPT also has an agreementwith experimen- A= 1 0 2− 0 2
RT
tal data and with this result in case when take into ac-
countcorrectionatO(p6)withq2 -dependence andelec- +(kp2)((p0p1)(p0p2)−p20(p1p2))
tromagnetic corrections [15]. Full agreement with this RT
resulthaveChPT incase ofO(p8)when electromagnetic +(kp0)((p0p2)(p1p2)−(p0p1)p22)), (A1)
corrections at O(e2) are included. The calculations in RT
NχQMcoincidewithladderDyson-Schwingercalculation
[44]whichalsotakeintoaccountthe dependenceofmass
on momentum of quark. One can say that nonperturba- (kp )((p p )(p p ) p2(p p ))
tioneffectsareincorporatedinsideNχQMbyinclusionof B = 1 0 2 0 1 − 0 1 2
RT
nonlocality structure of interaction quarks with mesons.
(kp )(p2p2 (p p )2)
Recently it was shown that picture of calculation will + 2 0 1− 0 1
change if one takes into account of intermediate vec- RT
tor meson resonance exchange [45, 46]. Similar picture +(kp0)((p0p1)(p1p2)−(p0p2)p21)), (A2)
shouldappearin the processesπγ∗ ππ/KK¯. We plan RT
→
toconsiderarolelightestvectormesoninthesereactions
inframeworkofextendedSU(2) SU(2)model[25,39].
×
It is also interesting in connection with future measure- (kp )((p p )(p p ) (p p )p2)
ment of exclusive γp π+π−p reaction of photopro- C = 1 0 2 1 2 − 0 1 2
→ RT
ductionatthe GlueXexperimentatJeffersonlaboratory
(kp )((p p )(p p ) p2(p p ))
[47]. + 2 0 1 1 2 − 1 0 2
In future we plan to calculate Dalitz decays of η and RT
η′ mesons into γπ+π− or e+e−π+π−. These decays are +(kp0)(p21p22−(p1p2)2)), (A3)
well-studied experimentally and will be measured in fu- RT
tureexperimentswithveryhighaccuracy. Theimportant
roleintheseprocessesplaysvectormesonswhichhaveto where denominator RT is
be included in extended SU(3) SU(3) model.
×
RT =p2p2p2+2(p p )(p p )(p p )
0 1 2 0 1 0 2 1 2
ACKNOWLEDGMENTS −p20(p1p2)2−(p0p1)2p22−(p0p2)2p21. (A4)
I thank A. E. Dorokhov, A. E. Radzhabov, These coefficients have a crossing symmetry: C =
D.V.KarlovetsandA.A.Shishmarevfordiscussionand A(p p ) and B =A(p p ).
0 1 0 2
↔− ↔−
[1] J. Steinberger, Phys. Rev.76, 1180 (1949). [7] A. G. Oganesian, A. V. Pimikov, N. G. Stefanis, and
[2] J. Wess and B. Zumino,Phys. Lett. 37B, 95 (1971). O. V. Teryaev,Phys. Rev.D93, 054040 (2016).
[3] E. Witten, Nucl.Phys. B 223, 422 (1983). [8] A. E. Dorokhov, A. E. Radzhabov, and A. Zhevlakov,
[4] E. Ruiz Arriola and W. Broniowski, Phys. Rev. D81, Eur. Phys. J. C71, 1702 (2011).
094021 (2010). [9] D. G. Dumm, S. Noguera, N. N. Scoccola, and
[5] M. Knecht and A. Nyffeler, Phys. Rev. D65, 073034 S. Scopetta, Phys. Rev.D89, 054031 (2014).
(2002). [10] A.M.BernsteinandB.R.Holstein,Rev.Mod.Phys.85,
[6] S. V. Mikhailov, A. V. Pimikov, and N. G. Stefanis, 49 (2013).
Phys.Rev.D82, 054020 (2010). [11] J. Bijnens, Nucl.Phys. B 367, 709 (1991).
7
[12] I. Giller, A. Ocherashvili, T. Ebertshauser, M. A. Phys. Lett. B417, 173 (1998).
Moinester, and S. Scherer, Eur. Phys. J. A25, 229 [30] L. Giusti, F. Rapuano, M. Talevi, and A. Vladikas,
(2005). Nucl. Phys. B538, 249 (1999).
[13] Yu.M. Antipovet al., Phys.Rev.D36, 21 (1987). [31] S. Aokiet al.,Eur. Phys.J. C74, 2890 (2014).
[14] G. Baum. et al, CERN-SPSLC96-14 (1996). [32] G. Colangelo et al.,Eur. Phys. J. C71, 1695 (2011).
[15] L. Ametller, M. Knecht, and P. Talavera, Phys. Rev. [33] J.-L. Kneur and A. Neveu,
D64, 094009 (2001). Phys. Rev.D92, 074027 (2015).
[16] I. V. Anikin, A. E. Dorokhov, and L. Tomio, Phys. [34] J. Terning, Phys. Rev.D44, 887 (1991).
Part. Nucl. 31, 509 (2000), [Fiz. Elem. Chast. Atom. [35] A. E. Dorokhov, A. E. Radzhabov, and A. Zhevlakov,
Yadra31,1023(2000)]. Eur. Phys. J. C75, 417 (2015).
[17] A. E. Dorokhov and L. Tomio, [36] S. Weinberg, Phys.Rev. 130, 776 (1963).
Phys.Rev.D62, 014016 (2000). [37] A. Salam, NuovoCim. 25, 224 (1962).
[18] S.Noguera, Int.J. Mod. Phys.E16, 97 (2007). [38] A.E.Dorokhov,A.E.Radzhabov, andA.S.Zhevlakov,
[19] S. Noguera and N. N. Scoccola, Eur. Phys. J. C72, 2227 (2012).
Phys.Rev.D78, 114002 (2008). [39] A. E. Dorokhov and W. Broniowski,
[20] A. Scarpettini, D. Gomez Dumm, and N. N. Scoccola, Eur. Phys. J. C32, 79 (2003).
Phys.Rev.D69, 114018 (2004). [40] A. Osipov, A. Radzhabov, and M. Volkov,
[21] A. E. Radzhabov and M. K. Volkov, Phys.Atom.Nucl. 70, 1931 (2007).
Eur. Phys.J. A19, 139 (2004), [Phys. Part. Nucl. [41] T. Hannah,Nucl. Phys.B593, 577 (2001).
Lett.1,1(2004)]. [42] T. N.Truong, Phys.Rev.D65, 056004 (2002).
[22] A. E. Radzhabov, D. Blaschke, M. Buballa, and M. K. [43] M. Hoferichter, B. Kubis, and D. Sakkas,
Volkov,Phys.Rev.D83, 116004 (2011). Phys. Rev.D86, 116009 (2012).
[23] D.Gomez Dumm,D. B. Blaschke, A.G. Grunfeld, and [44] S. R. Cotanch and P. Maris, Phys. Rev. D68, 036006
N.N. Scoccola, Phys. Rev.D73, 114019 (2006). (2003).
[24] We consider the isospin limit where masses of quarks u [45] R.A.Briceno,J.J.Dudek,R.G.Edwards,C.J.Shultz,
and d are equal. C. E. Thomas, andD.J. Wilson, Phys.Rev.Lett.115,
[25] M.IzzoVillafae, D.GomezDumm, andN.N.Scoccola, 242001 (2015).
Phys.Rev.D94, 054003 (2016). [46] R. A. Briceo, J. J. Dudek, R. G. Edwards, C. J. Shultz,
[26] I. Anikin, M. A. Ivanov, N. Kulimanova, and V. E. C. E. Thomas, and D. J. Wilson, Phys. Rev. D93,
Lyubovitskij, Phys. Atom. Nucl. 57, 1021 (1994), [Yad. 114508 (2016).
Fiz.57,1082(1994)]. [47] H. Al Ghoul et al. (GlueX), Proceedings, 16th Inter-
[27] D. Gomez Dumm, A. G. Grunfeld, and N. N. Scoccola, national Conference on Hadron Spectroscopy (Hadron
Phys.Rev.D74, 054026 (2006). 2015): Newport News, Virginia, USA, September 13-18,
[28] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, 2015, AIPConf. Proc. 1735, 020001 (2016).
Nucl.Phys. B147, 385 (1979).
[29] H. G. Dosch and S. Narison,
