Central exclusive diffractive production of $\pi^{+}\pi^{-}$ continuum, scalar and tensor resonances in $pp$ and $p \bar{p}$ scattering within tensor pomeron approach

2 MB·

by  Piotr Lebiedowicz

Central exclusive diffractive production
of π+π continuum, scalar and tensor resonances
−
in pp and pp¯ scattering within tensor pomeron approach
Piotr Lebiedowicz,1, Otto Nachtmann,2,† and Antoni Szczurek ‡1,§
∗
1Institute of Nuclear Physics, Polish Academy of Sciences,
Radzikowskiego 152, PL-31-342 Kraków, Poland
2Institut für Theoretische Physik, Universität Heidelberg,
6 Philosophenweg 16, D-69120 Heidelberg, Germany
1
0 Abstract
2
We consider central exclusive diffractive dipion production in the reactions pp ppπ+π and
−
n →
pp¯ pp¯π+π athigh energies. Weinclude thedipioncontinuum,thedominant scalar f (500),
a − 0
→
J f (980),andtensor f (1270)resonancesdecayingintotheπ+π pairs. Thecalculationisbasedon
0 2 −
8
atensorpomeronmodelandtheamplitudesfortheprocessesareformulatedintermsofvertices
1
respectingthestandardcrossingandcharge-conjugationrelationsofQuantumFieldTheory. The

] formulaeforthedipioncontinuumandtensormesonproductionaregivenhereforthefirsttime.
h
p The theoretical results are compared with existing STAR, CDF, CMS experimental data and pre-
-
p dictions for planned or being carried out experiments(ALICE, ATLAS) are presented. We show
e the influence of the experimentalcuts on the integratedcross section and on various differential
h
[ distributionsforoutgoingparticles. Distributionsinrapiditiesandtransversemomentaofoutgo-

ingprotonsandpionsaswellascorrelationsinazimuthalanglebetweenthemarepresented. We
1
v find that the relative contribution of resonant f (1270) and dipion continuum strongly depends
2
7
on the cut on proton transverse momenta or four-momentum transfer squared t which may
3 1,2
5 explain somecontroversialobservationsmadebydifferent ISRexperimentsin thepast. Thecuts
4
may play then the role of a ππ resonance filter. We suggest some experimental analyses to fix
0
. modelparametersrelatedtothepomeron-pomeron-f2 coupling.
1
0
6 PACSnumbers: 12.40.Nn,13.60.Le,14.40.Be
1
:
v
i
X
r
a
‡ AlsoatUniversityofRzeszów,PL-35-959Rzeszów,Poland.
Electronicaddress:[email protected]
∗
†Electronicaddress:[email protected]
§Electronicaddress:[email protected]
1
I. INTRODUCTION
The exclusive reaction pp ppπ+π is one of the reactions being extensively stud-
−
→
ied by several experimental groups such as COMPASS [1–3], STAR [4, 5], CDF [6, 7],
ALICE [8], ATLAS [9], and CMS [10, 11]. It is commonly believed that at high ener-
gies the pomeron-pomeron fusion is the dominant mechanism of the exclusive two-pion
production. In the past two of us have formulated a simple Regge inspired model of
the two-pion continuum mediated by the double pomeron/reggeon exchanges with pa-
rameters fixed from phenomenological analyses of NN and πN scattering [12] 1. The
numberoffreemodelparametersisthenlimitedtoaparameterofformfactordescribing
off-shellness oftheexchanged pion. Thelargest uncertaintiesin themodel areduetothe
unknown off-shell pion form factor and the absorption corrections discussed recently in
[16]. Although this model gives correct order of magnitude cross sections it is not able
to describe detailsof differential distributions, in particular the distribution in dipion in-
variant mass where we observe a rich pattern of structures. Clearly such an approach
doesnotincluderesonancecontributions whichinterferewiththecontinuum. Itisfound
that the pattern of visible structures depends on experiment but as we rather advocate
on the cuts used in a particular experiment (usually these cuts are different for different
experiments).
It was known for a long time that the commonly used vector pomeron has problems
fromafieldtheorypointofview. Takenliterallyitgivesoppositesignsfor ppand p¯ptotal
cross sections. Awayoutofthisdilemmawasalreadyshown in[17]wherethe pomeron
was described as a coherent superposition of exchanges with spin 2 + 4 + 6 + ... . This
same idea is realised in a very practical way in the tensor-pomeron model formulated
in [18]. In this model pomeron exchange can effectively be treated as the exchange of
a rank-2 tensor. The corresponding couplings of the tensorial object to proton and pion
were worked out. In Ref. [19] the model was applied to the production of several scalar
and pseudoscalar mesons in the reaction pp ppM. A good description of the experi-
→
mentaldistributions [20]wasachievedatrelativelylowenergywhere,however,reggeon
exchangesstillplayaveryimportantrole2. Theresonant(ρ0 π+π )andnon-resonant
−
→
(Drell-Söding) photon-pomeron/reggeon π+π production in pp collisions wasstudied
−
in [22]. In [23] an extensive study of the photoproduction reaction γp π+π p in the
−
→
framework of the tensor-pomeron model waspresented.
In most of the experimental preliminary spectra of the pp ppπ+π reaction at
−
→
higher energies a peak at M 1270 MeV is observed. One can expect that the peak
ππ
∼
is related to the production of the well known tensor isoscalar meson f (1270) which
2
decays with high probability into the π+π channel. In principle, contributions from
−
the f (1370), f (1500) and f (1710) mesons are not excluded. The f (1500) and f (1710)
0 0 0 0 0
mesonsareoftenconsideredaspotentialcandidatesforscalarstateswithdominantglue-
ball content and it is expected that in pomeron-pomeron fusion the glueball production
could be prominently enhanced dueto the gluonic nature ofthe pomeron [24, 25].
For a study of the resonance production observed in the π+π and K+K massspec-
− −
1Foranotherrelatedworksee[13]wheretheexclusivereaction pp ppπ+π constitutesanirreducible
−
→
background to the scalar χ meson production. These model studieswere extendedalso tothe pp
c0
→
ppK+K [14]andthe pp nnπ+π+ [15]processes.
−
→
2TheroleofsecondaryreggeonsincentralpseudoscalarmesonproductionwasdiscussedalsoinRef.[21].
2
tra in the fixed target experiments at low energies see Refs. [26–30]. There is evidence
from theanalysisofthedecaymodesofthescalarstatesobserved,thatthelightestscalar
glueball manifests itself through the mixing with nearby qq¯ states [30] and that the dif-
ference in the transverse momentum vectors between the two exchange particles (dP)
t
can be used to select out known qq¯ states from non-qq¯ candidates. 3 Also the four-
momentumtransfersquared t fromoneoftheprotonverticesfortheresonanceswasde-
| |
termined. For the tensor f (1270) and f (1525) states their fractional distributions show
2 2′
non-single exponential behaviour; see Fig. 5 of [27]. It has been observed in [26] that the
ρ(770), φ(1020), f (1270) and f (1525) resonances arevisible more efficiently inthe high
2 2′
t = t +t region (t > 0.3 GeV2)and thatatlow t their signalsare suppressed.
1 2
| |
In the ISR experiments, see [32–34], the π+π invariant mass distribution shows an
−
enhancement in the low-mass (S-wave) region and a very significant resonance struc-
ture. A clear f (1270) signal has been observed at √s = 62 GeV [32] and a cross section
2
σ(pp ppf , f π+π )of(8 1 3)µbwasdetermined,wherethefour-momentum
2 2 −
→ → ± ±
transfer squared is t > 0.08 GeV2, the scattered protons have x > 0.9, and the pion
F,p
| |
c.m.systemrapidityislimitedtotheregion y > 1.5. However,thisbehaviourisrather
π
| |
different from that observed in Ref. [35]. In our opinion this is due to the different kine-
matic coverage of these two ISR experiments. The experiment [35] has been performed
at √s = 63 GeV. Compared to [32]their analysis covered smallerfour-momentum trans-
fer squared (0.01 . t . 0.06 GeV2), x > 0.95, and the central rapidity region was
F,p
| |
more restricted. Moreover, their D-wave cross section shows an enhancement between
1.2 1.5GeV and the authors of [35]argued that the f (1270) alone does notexplain the
2
−
behaviour of the data and additional states are needed, e.g. a scalar at around 1400 MeV
andtensoratm = 1480 50MeVwith Γ = 150 50MeV.Inthe paper[34]thecross sec-
± ±
tion ofcentral π+π production showsanenhancementinboth the S and D-wavesnear
−
the mass of the f (1270) and f (1400). The D-wave mass spectrum was described with
2 0
the f (1270) resonant state and a broad background term. A cross section for exclusive
2
f (1270) meson production of5.0 0.7 µb(notincludingasystematic error, estimated to
2
±
be 1.5 µb) wasobtained.
On the theoretical side, the production of the tensor meson f was not considered so
2
farintheliterature,exceptinRef.[36]. Wenotethatinarecentwork[37]theauthorsalso
considertheresonanceproduction through thepomeron-pomeron fusionattheLHCbut
ignoring the spin effects in the pomeron-pomeron-meson vertices. In the present pa-
per we consider both, production of the two-pion continuum and of the f (500), the
0
f (980),andthe f (1270)resonancesintheπ+π channel,consistentlywithinthetensor
0 2 −
pomeronmodel. Thismodelallowsalsotocalculateinterferenceeffects. Weconsiderthe
tensor-tensor-tensor coupling in a Lagrangian formalism and present a list of possible
couplings. The specificities of the different couplings are discussed, also in the context
of experimental results. We discuss a first qualitative attempt to “reproduce” the exper-
imentally observed behaviours of the two-pion spectra obtained in the pp ppπ+π
−
→
reaction and discuss consequences of experimental cuts on the observed spectra. The
calculations presented in section V were done with a FORTRAN code using the VEGAS
3IthasbeenobservedinRef.[31]thatalltheundisputedqq¯states(i.e. η,η , f (1285)etc.) aresuppressed
′ 1
as dP 0, whereas the glueball candidates, e.g. f (1500), survive. As can be seen there ρ0(770),
t 0
→
f (1270)and f (1525)havelargerdP andtheircrosssectionspeakatφ =π,i.e. theoutgoingprotons
2 2′ t pp
areonoppositesidesofthebeam,incontrasttothe’enigmatic’ f (980), f (1500)and f (1710)states.
0 0 0
3
routine [38].
II. EXCLUSIVETWO-PIONPRODUCTION
We study central exclusive production of π+π in proton-proton collisions at high
−
energies
p(p ,λ )+ p(p ,λ ) p(p ,λ )+π+(p )+π (p )+ p(p ,λ ), (2.1)
a a b b 1 1 3 − 4 2 2
→
where p , p and λ , λ +1/2, 1/2 denote the four-momenta andhelicities of
a,b 1,2 a,b 1,2
∈ { − }
the protons, and p denote the four-momenta of the charged pions, respectively.
3,4
The full amplitude of π+π production is a sum of continuum amplitude and the
−
amplitudesthrough the s-channel resonances:
Mpp→ppπ+π− = Mπppπ→−pcopnπt+inπu−um+Mπppπ→−prepsπo+naπn−ces. (2.2)
This amplitude for central exclusive π+π production is believed to be given by the fu-
−
sionoftwoexchangeobjects. Thegeneric“Bornlevel”diagramisshown inFig.1,where
welabeltheexchangeobjectsbytheirchargeconjugationnumbersC ,C +1, 1 . At
1 2
∈ { −P}
highenergiestheexchangeobjectstobeconsideredarethephotonγ,thepomeron ,the
odderon O, and the reggeons R = f2R,a2R,ωR,ρR. Their charge conjugation numbers
and G paritiesare listed in Table I.
Incalculatingtheamplitude(2.2)fromthediagramFig.1asumoverallcombinations
ofexchanges, (C ,C ) = (1,1), ( 1, 1), (1, 1), ( 1,1), hasto be taken:
1 2
− − − −
= (1,1) + ( 1, 1)+ (1, 1)+ ( 1,1). (2.3)
pp ppπ+π − − − −
M → − M M M M
Note that the (1,1) and ( 1, 1) contributions will produce a π+π state with charge
−
− −
conjugation C = +1. The (1, 1) and ( 1,1) contributions will produce a π+π state
−
− −
with C = 1. This implies that the (C1,C2) amplitudes have the following properties
− M
under exchange of the π+ and π momenta, keeping all other kinematic variables, indi-
−
p(pa) p(p1)
C1
π+(p3)
π−(p4)
C2
p(pb) p(p2)
FIG. 1: Generic “Born level” diagram for central exclusive π+π production in proton-proton
−
collisions.
4
TABLEI: Chargeconjugationand G-parityquantumnumbersofexchangeobjectsforresonance
andcontinuumproduction.
Exchangeobject C G
P 1 1
f2R 1 1
a2R 1 -1
γ -1
O -1 -1
ωR -1 -1
ρR -1 1
cated bythe dots, fixed:
(1,1)(...,p ,p ) = (1,1)(...,p ,p ),
3 4 4 3
M M
( 1, 1)(...,p ,p ) = ( 1, 1)(...,p ,p ),
− − 3 4 − − 4 3
M M (2.4)
(1, 1)(...,p ,p ) = (1, 1)(...,p ,p ),
− 3 4 − 4 3
M −M
( 1,1)(...,p ,p ) = ( 1,1)(...,p ,p ).
− 3 4 − 4 3
M −M
The interference of the amplitudes (1,1) + ( 1, 1) with (1, 1) + ( 1,1) will
− − − −
M M M M
leadtoasymmetriesunderexchangeoftheπ+ andπ momenta. Suchasymmetriescan,
−
forinstance, bestudiedintherest frameofthe π+π pairusingaconvenientcoordinate
−
systemliketheCollins-Soperframe[39]. Foradiscussion ofvariousreferenceframessee
for instance [23, 40]. Asymmetries may be quite interesting from an experimental point
of view since they could allow to measure small contributions in the amplitude which
wouldbehard todetectotherwise. Somedetailsrelated fortheasymmetriesaregivenin
AppendixB.
III. TWO-PIONCONTINUUMPRODUCTION
The generic diagrams for exclusive two-pion continuum production are shown in
Fig. 2. Taking into account the G parity of -1 for the pions we get the following com-
p(pa) p(p1) p(pa) p(p1)
t1 t1
γ,IP,IR γ,IP,IR
π+(p3) π−(p4)
tˆ uˆ
π−(p4) π+(p3)
γ,IP,IR γ,IP,IR
p(pb) t2 p(p2) p(pb) t2 p(p2)
FIG.2: TheBorndiagrams fordouble-pomeron/reggeonand photonmediatedcentralexclusive
continuumπ+π productioninproton-protoncollisions.
−
5
binations (C ,C ) ofexchanges which can contribute 4
1 2
(C1,C2) = (1,1) : (P+ f2R,P+ f2R); (3.1)
(C1,C2) = ( 1, 1) : (ρR +γ,ρR +γ); (3.2)
− −
(C1,C2) = (1, 1) : (P+ f2R,ρR +γ); (3.3)
−
(C1,C2) = ( 1,1) : (ρR +γ,P+ f2R). (3.4)
−
Note that for the cases involving the photon γ in (3.2) to (3.4) one also has to take into
account the diagrams involving the corresponding contact terms; see[23].
From the above list of exchange contributions we have already treated (P + f2R,γ)
and (γ,P + f2R) in [22]. At high energies the contributions involving ρR exchanges are
expectedtobesmallsinceρR issecondaryreggeonanditscouplingtotheprotonissmall;
see e.g. (3.61), (3.62) of [18]. The (γ,γ) contribution is higher order in α . Thus we are
em
leftwith the contribution (P+ f2R,P+ f2R), (3.1), which weshall treat now.
TheamplitudeforthecorrespondingdiagramsinFig.2canbewrittenasthefollowing
sum: 5
Mπppπ→−pcopnπt+inπu−um = M(PP→π+π−)+M(Pf2R→π+π−) +M(f2RP→π+π−) +M(f2Rf2R→π+π−).
(3.5)
PP
The -exchange amplitudecan be written as
(PP π+π−) = (ˆt) + (uˆ) , (3.6)
M → Mλaλb→λ1λ2π+π− Mλaλb→λ1λ2π+π−
where
(ˆt)
=
Mλaλb→λ1λ2π+π−
( i)u¯(p ,λ )iΓ(Ppp)(p ,p )u(p ,λ )i∆(P)µ1ν1,α1β1(s ,t )iΓ(Pππ)(p , p )i∆(π)(p )
− 1 1 µ1ν1 1 a a a 13 1 α1β1 t − 3 t
iΓ(Pππ)(p ,p )i∆(P)α2β2,µ2ν2(s ,t )u¯(p ,λ )iΓ(Ppp)(p ,p )u(p ,λ ),
× α2β2 4 t 24 2 2 2 µ2ν2 2 b b b
(3.7)
(uˆ)
=
Mλaλb→λ1λ2π+π−
( i)u¯(p ,λ )iΓ(Ppp)(p ,p )u(p ,λ )i∆(P)µ1ν1,α1β1(s ,t )iΓ(Pππ)(p ,p )i∆(π)(p )
− 1 1 µ1ν1 1 a a a 14 1 α1β1 4 u u
iΓ(Pππ)(p , p )i∆(P)α2β2,µ2ν2(s ,t )u¯(p ,λ )iΓ(Ppp)(p ,p )u(p ,λ ),
× α2β2 u − 3 23 2 2 2 µ2ν2 2 b b b
(3.8)
4NotethatGparityinvarianceforbidstheverticesa2Rππ,ωRππ,Oππ,seeTableI.Thus,theexchanges
ofa2R,ωR,Ocannotcontributetothedipioncontinuum.
5We emphasize, that not only the leading pomeron exchanges contribute to the dipion system with the
isospin I = 0 and C = +1 but also the Pf2R, f2RP, f2Rf2R, ρRρR exchanges and due to their non-
negligibleinterferenceeffectswiththeleadingPPtermthesubleading f2R exchangesmustbeincluded
explicitlyinourcalculations.
6
where p = p p p and p = p p + p , s = (p + p )2. The kinematic variables
t a 1 3 u 4 a 1 ij i j
− − −
for reaction (2.1)are
s = (p + p )2 = (p + p + p + p )2, s = M2 = (p + p )2,
a b 1 2 3 4 34 ππ 3 4
t = q2, t = q2, q = p p , q = p p . (3.9)
1 1 2 2 1 a − 1 2 b − 2
Here ∆(P) and Γ(Ppp) denote theeffective propagator andproton vertexfunction, respec-
tively, for the tensorial pomeron. For the explicit expressions, see Sec. 3 of [18]. The
normal pion propagator is i∆(π)(k) = i/(k2 m2π). In a similar way the Pf2R, f2RP and
−
f2Rf2R amplitudescan be written.
Thepropagator of the tensor-pomeron exchange iswritten as(see Eq.(3.10)of [18]):
i∆µ(Pν,)κλ(s,t) = 41s gµκgνλ +gµλgνκ − 12gµνgκλ (−isαP′ )αP(t)−1 (3.10)
(cid:18) (cid:19)
and fulfilsthe following relations
∆(P) (s,t) = ∆(P) (s,t) = ∆(P) (s,t) = ∆(P) (s,t),
µν,κλ νµ,κλ µν,λκ κλ,µν
(3.11)
gµν∆(P) (s,t) = 0, gκλ∆(P) (s,t) = 0.
µν,κλ µν,κλ
For the f2R reggeon exchange a similar form of the effective propagator and the f2Rpp
and f2Rππ effective vertices is assumed, see (3.12) and (3.49), (3.53) of [18]. Here the
pomeron and reggeon trajectories α (t), where i = P,R, are assumed to be of standard
i
linearforms, seee.g. [41],
αP(t) = αP(0)+αP′ t, αP(0) = 1.0808, αP′ = 0.25 GeV−2, (3.12)
αR(t) = αR(0)+αR′ t, αR(0) = 0.5475, αR′ = 0.9 GeV−2. (3.13)
The corresponding coupling of tensor pomeron to protons (antiprotons) including a
vertex form-factor, iswritten as(see Eq.(3.43)of[18]):
iΓ(Ppp)(p ,p) = iΓ(Pp¯p¯)(p ,p)
µν ′ µν ′
1 1
= i3βPNNF1 (p′ p)2 γµ(p′ + p)ν +γν(p′ + p)µ gµν(p/′ + p/) , (3.14)
− − 2 − 4
(cid:26) (cid:27)
(cid:0) (cid:1) (cid:2) (cid:3)
where βPNN = 1.87 GeV−1. Starting with the Pππ coupling Lagrangian, see Eq. (7.3) of
P
[18], we havethe following ππ vertex (see Eq.(3.45)of[18])
iΓµ(Pνππ)(k′,k) = i2βPππ (k′ +k)µ(k′ +k)ν 1gµν(k′ +k)2 FM((k′ k)2), (3.15)
− − 4 −
(cid:20) (cid:21)
where βPππ = 1.76 GeV−1 gives proper phenomenological normalization. The form
factors taking into account that the hadrons are extended objects (see Section 3.2 of [41])
are chosen as
4m2 2.79t 1
F (t) = p − , F (t) = , (3.16)
1 (4m2 t)(1 t/m2 )2 M 1 t/Λ2
p − − D − 0
7
where m is the proton mass and m2 = 0.71 GeV2 is the dipole mass squared and
p D
Λ2 = 0.5GeV2;seeEq.(3.34)of[18]. Alternatively,insteadoftheproductoftheformfac-
0
tors F (t)F (t) where thetwo factors are attached to the relevantvertices (seeEqs. (3.14)
1 M
(P/R)
and (3.15)) we can take single form factors F (t) in the exponential form where the
πN
pomeron/reggeon slopeparametershavebeenestimatedfromafittotheπp elasticscat-
tering data, seeEq. (2.11)and Fig. 3 of [16].
The amplitudes (3.7) and (3.8) must be “corrected” for the off-shellness of the inter-
mediate pions. The form of the off-shell pion form factor is unknown in particular at
higher values of p2 or p2. The form factors are normalized to unity at the on-shell point
t u
Fˆ (m2) = 1and parametrised here in two ways:
π π
k2 m2
Fˆ (k2) = exp − π , (3.17)
π
Λ2
!
off,E
Λ2 m2
Fˆ (k2) = off,M− π , (3.18)
π
Λ2 k2
off,M−
where Λ2 or Λ2 could be adjusted to experimental data. It was shown in Fig. 9
off,E off,M
of [16] that the monopole form (3.18) is supported by the preliminary CDF results [7]
>
particularly at higher values of two-pion invariant mass, M 1.5 GeV. Thus, in the
ππ
numerical calculations below, see section V, we used the monopole form of the off-shell
pion form factors.
Inthe high-energy small-angle approximation wehave
u¯(p ,λ )γ (p + p) u(p,λ) (p + p) (p + p) δ (3.19)
′ ′ µ ′ ν ′ µ ′ ν λλ
→ ′
and wecan write the leadingtermsof the amplitudesforthe pp ppπ+π process as
−
→
(ˆt)
Mλaλb→λ1λ2π+π− ≃ 3βPNN2(p1 + pa)µ1(p1 + pa)ν1 δλ1λa F1(t1)FM(t1)
1 [Fˆ (p2)]2
×2βPππ(pt − p3)µ1(pt − p3)ν1 4s (−is13αP′ )αP(t1)−1 p2π tm2
13 t − π (3.20)
1
2βPππ(p4 + pt)µ2(p4 + pt)ν2 ( is24αP′ )αP(t2)−1
× 4s −
24
×3βPNN2(p2 + pb)µ2(p2 + pb)ν2 δλ2λb F1(t2)FM(t2),
(uˆ)
Mλaλb→λ1λ2π+π− ≃ 3βPNN2(p1 + pa)µ1(p1 + pa)ν1 δλ1λa F1(t1)FM(t1)
1 [Fˆ (p2)]2
×2βPππ(p4 + pu)µ1(p4 + pu)ν1 4s (−is14αP′ )αP(t1)−1 p2π um2
14 u π (3.21)
−
1
2βPππ(pu p3)µ2(pu p3)ν2 ( is23αP′ )αP(t2)−1
× − − 4s −
23
×3βPNN2(p2 + pb)µ2(p2 + pb)ν2 δλ2λb F1(t2)FM(t2).
Now,weconsiderthevectorpomeron exchangemodel. Wehavethefollowingansatz
P
for the π π vertex omitting the form factors (M 1 GeV)
V − − 0
≡
iΓµ(PVπ−π−)(k′,k) = i2βPππM0(k′ +k)µ. (3.22)
−
8
From isospin andcharge-conjugation invariance we should have
iΓ(PVπ+π+)(k ,k) = iΓ(PVπ−π−)(k ,k). (3.23)
µ ′ µ ′
Butthe crossing relations require
iΓ(PVπ+π+)(k ,k) = iΓ(PVπ−π−)( k, k ). (3.24)
µ ′ µ ′
− −
Andfrom (3.22)we get
iΓ(PVπ−π−)( k, k ) = iΓ(PVπ−π−)(k ,k). (3.25)
µ ′ µ ′
− − −
Clearly, (3.23) and (3.24) plus (3.25) would lead to iΓ(PVππ)(k ,k) 0. This is another
µ ′
P ≡
manifestation that the ππ coupling for a vector pomeron hasbasicproblems; see also
V
the discussion in Sec. 6.1 of[18].
IV. DIPIONRESONANTPRODUCTION
In this section we consider the production of s-channel resonances which decay to
π+π
−
p+ p p+(resonance π+π )+ p. (4.1)
−
→ →
Theresonanceswhichshouldbetakenintoaccounthereandtheirproduction modesvia
(C ,C ) fusion are listed in TableII.
1 2
The production of ρ(770) and ρ(1450) was already treated in [22]. Here we shall dis-
cuss the production of the f and f resonances; see Fig. 3. We shall concentrate on the
0 2
TABLEII: Resonancesand(C ,C )productionmodes.
1 2
IGJPC resonance production(C ,C )
1 2
0+0++ f0(500) (P+ f2R,P+ f2R),(a2R,a2R),
f0(980) (O+ωR+γ,O+ωR+γ),(ρR,ρR),
f0(1370) (γ,ρR),(ρR,γ)
f (1500)
0
f (1710)
0
1+1−− ρ(770) (γ+ρR,P+ f2R),(P+ f2R,γ+ρR),
ρ(1450) (O+ωR,a2R),(a2R,O+ωR)
ρ(1700)
0+2++ f2(1270) (P+ f2R,P+ f2R),(a2R,a2R),
f2′(1525) (O+ωR+γ,O+ωR+γ),(ρR,ρR),
f2(1950) (γ,ρR),(ρR,γ)
1+3−− ρ3(1690) (γ+ρR,P+ f2R),(P+ f2R,γ+ρR),
(O+ωR,a2R),(a2R,O+ωR)
0+4++ f4(2050) (P+ f2R,P+ f2R),(a2R,a2R),
(O+ωR+γ,O+ωR+γ),(ρR,ρR),
(γ,ρR),(ρR,γ)
9
p(p ) p(p )
a 1
γ,IP,IR π+(p3)
f ,f
0 2
γ,IP,IR π (p )
− 4
p(p ) p(p )
b 2
FIG. 3: The Born diagram for double-pomeron/reggeonand photon mediated central exclusive
IGJPC = 0+0++ and 0+2++ resonances production and their subsequent decays into π+π in
−
proton-protoncollisions.
contributions from (C1,C2) = (P+ f2R,P+ f2R). Wecan justify this asfollows. The con-
tributions involving the odderon (if it exists at all), the a2R and the ρR should be small
duetosmallcouplingsoftheseobjectstotheproton. Thesecondaryreggeonsa2R,ωR,ρR
should give small contributions at high energies. We shall neglect contributions involv-
ingthe photon γ in the following. Theseare expected tobecome important onlyfor very
small values of t1 and/or t2 . Thus we are left with (P + f2R,P + f2R) where (P,P)
fusion is the lead|in|g term an|d|(P, f2R) plus (f2R,P) is the first non-leading term due to
reggeons at high energies.
Theamplitudeforexclusive resonant π+π production, givenbythediagram shown
−
in Fig. 3, can be written as
Mπppπ→−prepsπo+naπn−ces = M(λPaλPb→→fλ01→λ2ππ++ππ−−) +M(λPaλPb→→fλ21→λ2ππ++ππ−−). (4.2)
A. IGJPC = 0+0++
For ascalar meson, JPC = 0++,the amplitude for PP fusion can be written as
M(λPaλPb→→fλ01→λ2ππ++ππ−−) =(−i)u¯(p1,λ1)iΓµ(P1νp1p)(p1,pa)u(pa,λa) i∆(P)µ1ν1,α1β1(s1,t1)
iΓ(PPf0) (q ,q ) i∆(f0)(p ) iΓ(f0ππ)(p ) (4.3)
× α1β1,α2β2 1 2 34 34
i∆(P)α2β2,µ2ν2(s ,t ) u¯(p ,λ )iΓ(Ppp)(p ,p )u(p ,λ ),
× 2 2 2 2 µ2ν2 2 b b b
where s = (p +q )2 = (p + p )2, s = (p +q )2 = (p + p )2, and p = p + p .
1 a 2 1 34 2 b 1 2 34 34 3 4
PP
The effective Lagrangians andthe vertices for fusion into the f meson are discussed
0
PP
in AppendixA of [19]. Aswas shown there the tensorial f vertex corresponds to the
0
sumoftwolowestvaluesof(l,S), thatis(l,S) = (0,0) and(2,2)withthecorresponding
coupling parameters g and g , respectively. The vertex including a form factor
P′ PM P′′PM
readsthen asfollows (p = q +q )
34 1 2
iΓ(µPν,Pκλf0)(q1,q2) = iΓ′µ(νP,κPλf0) |bare +iΓ′µ′ν(P,κPλf0)(q1,q2) |bare F˜(PPf0)(q21,q22,p234); (4.4)
(cid:16) (cid:17)
10