A model of a transition neutral pion formfactor measured in annihilation and scattering channels

0.14 MB·English

by  E. A. Kuraev

A model of a transition neutral pion formfactor measured in annihilation and
scattering channels
Yu. M. Bystritskiy,∗ V. V. Bytev,† and E. A. Kuraev‡
JINR-BLTP, 141980 Dubna, Moscow region, Russian Federation
A. N. Ilyichev§
National Scientific and Educational Centre of Particle and High Energy
Physics of the Belarusian State University, 220040 Minsk, Belarus
We consider an alternative explanation of newly found growth of neutral pion transition form
factor with virtualityof oneofphoton. It isbased on Sudakovsuppression of quark-photonvertex.
Some applications to scattering and annihilation channels are considered including the relevant
0 experimentswith lepton-proton scattering.
1
0 PACSnumbers: 13.60.-r,13.66.Bc
2

n
a I. INTRODUCTION e–(p– ) e –(p–′ )
J

1 Alotofattentionwaspaidtotheproblemofdescribing
2 the transition form factor of neutral pion [1–3]. It is the γ (q1) π (q )
π
informationaboutwavefunctionofneutralpion,namely
]
h the distribution on the energy fractions of u,d quarks
p inside a neutralpion, is the motivationof numerous the- γ*(q)
- oretical approaches to describe the transition form fac-
p
tor. Recently some experimental information about it’s
he behavior was obtained in process e+e− → e+e−π0. The e +(p+ ) e +(p+′ )
[ kinematics, when one of photon is almost real and the
Fig. 1: Scattering process of pion production
other is highly virtual was considered [4, 5]. The result
4
was presented by authors of the experiment as a some
v
8 nondecreasing function of the module of square of mo- (p2 = p′2 = m2, q2 = M2) neutral pion is created by
± ± e π
6 mentum of a virtual photon. Such type of behavior is in twophotonswithmomentaq =p −p′ andq =p −p′
6 clear contradiction with the predictions of factorization that involving into lepton intera+ction+as it i1s pres−ente−d
3 theoremapplyingtothisprocess(see[6–8]andreferences
onFig.1. DuetoWeizs¨acker-Williams(WW)kinematics
.
2 therein). of this process (the scattered electron assumed to move
1 Belowweconsideranotherreasontoexplainsuchtype
close to beam direction) one of photons is almost real
9 of behavior, using the well known expression of a vir- q2 ≪M2 and other is off mass shell Q2 =−q2 ≫M2.
0 1
tual photon-quark vertex (so called Sudakov form factor
Matrix element has a form (see for details in the Ap-
:
v [9] ) which is entered in the triangle Feynman diagram, p(cid:12)(cid:12)en(cid:12)(cid:12)dix A)
i describing the conversion of two photon to the neutral
X
pseudoscalar meson. It is the motivation of this paper. 2s(4πα)2
ar Both channels of pseudoscalar mesons production in M = q2q12 [q×q1]zV(Q2)N+N−,
elasticelectron-positroncollisions-thescatteringandan- M2 Q2
nihilation ones are considered. The second one e+e− → V(Q2) = q F , (2)
π0l+l− can be the subject of experimental investigation. 2π2FπQ2 (cid:18)Mq2(cid:19)
1
|N |2 = Tr[p p p p ]=2,
± s2 − + − +
II. SCATTERING CHANNEL
where s=(p +p )2 ≫Q2 and V(Q2) is the transition
+ −
form-factorofpion,F =93MeVisthedecayconstantof
π
In the scattering type of experiments pion, M = M = M = 280 MeV is the quark mass, q,
q u d
e+(p )+e−(p )→e+(p′ )e−(p′ )π0(q ), (1) q1, are transversal to the beam axes (z) direction com-
+ − + − π ponents of photon momenta. The quantity F(Q2/M2)
q
has a form:
d4k
F Q2/M2 =− ×
∗Electronicaddress: [email protected] q iπ2
†Electronicaddress: [email protected] Z
(cid:0) (cid:1) Q2V (Q2,p2,p2)
‡Electronicaddress: [email protected] × S 1 2 ,(3)
§Electronicaddress: [email protected] (k2−M2+i0)(p2−M2+i0)(p2−M2+i0)
q 1 q 2 q
2
γ (q1) for the cross section:
dσ α4
= V2(Q2)J(Q2),
dQ2 4Q2
q(k)
1
J(Q2) = L2+L (L −1)−(L +1), (9)
2 s s e e
π (q )
q(k+q) π
1 where
s Q2
q(k+qπ) Ls =lnQ2+M2, Le =lnm2. (10)
e
γ*(q) Thereareseveralapproachestoinferthe valueV(Q2),
which is named as pion transition formfactor.
Fig. 2: Triangle vertex for γγ∗ →π0 process One of them is based on QCD collinear factorization
theorem [2]
1
where p1 =k+q1, and p2 =k+qπ and Sudakov vertex VBL(Q2)= 2Fπ dx φ (x,s). (11)
function VS [10, 11] is: 3 xQ2 π
Z
0
α C Q2 Q2
VS(Q2,p21,p22)=exp − s2πF ln|p2|ln|p2| , (4) andinthepapers[6,8]differentformsofpionwavefunc-
(cid:18) 1 2 (cid:19) tion φ (x,s) was used. Another possible mechanisms of
π
the effect was given in [7, 12].
where Q2 ≫ |p2 | ≫ M2 and C = N2−1 /(2N) =
1,2 q F Also in the paper [13],[14] was pointed that pion form
4/3. We use here the the Goldberger-Treiman relation
(cid:0) (cid:1) factorintheframesconstituentquarkmodelhasthedou-
on the quark level F =M /g =93 MeV.
π q qq¯π ble logarithmic asymptotic at large momentum transfer.
The cross section of the process (1) has a form:
Another one which use the approachof Nambu–Jona-
Lasinio model [15, 16] gives:
1
dσ = |M|2dΓ (5)
3
8s 2F
VNJL(Q2)= π. (12)
3Q2
ThephasevolumeofthefinalstatedΓ canbeexpressed
3
throughtheSudakovparametrizationofthephoton’smo-
The approachused hereis basedonthe Sudakovformof
menta which turns out to be convenient:
thevertexfunctionwhichdescribeinteractionofaphoton
with large four momentum square |q2| with two quarks
q = α p˜ +β p˜ +q ,
1 1 + 1 − 1⊥ of an anomalous three angles quark diagram describing
q = αp˜++βp˜−+q⊥, (6) the conversionof two photons to the neutral pion.
a p = 0, q2 =−q2 , The three angle quark-loop diagramm itself at large
⊥ ± 1⊥ 1⊥
Q2 has the double-logarithm asymptotic [14] and inser-
and we imply the 4-vectors p˜ to be light-like, 2p˜ p˜ = tion of Sudakov form of the vertex function which in-
± + −
s. Therefore (details in Appendix A): cludes also QCD-inspired corrections gives the asymp-
totic ln ln Q2/m2 behavior.
dΓ = (2π)−5δ4 p +p −p′ −p′ −q × The final expression is (details are in Appendix A):
3 + − + − π (cid:0) (cid:0) (cid:1)(cid:1)
d3p′ d3p′ q3q
× 2E++′ 2E−′−(cid:0)2Eππ = (cid:1) V(Q2)=A2πFMαq2C Φ(zB), (13)
π s F
1 dβ
= (2π)−5 1 d2q1d2q, (7) where
4sβ (1−β )
1 1
Q2+M2 1
<β <1. dx
s 1 Φ(z) = 1−e−zBx(1−x) ,
x
Using the expressionfor the squareofmomentumof”al- Z0 (cid:16) (cid:17)
C α Q2
most” real photon z = F s ln2 ,
B 2π BM2
q
1
q2 =− q2 +m2β2 , q2 ≪Q2, (8)
1 1−β 1⊥ e 1 1⊥ whereA,B canbeconsideredasapositivefittingparam-
1
(cid:2) (cid:3) eters of order of unity. We find it through Babar data
andperformingtheintegrationontheparametersofscat- fitting. Function Q2V(Q2) is presented in Fig. 3, where
tered electron (β ,~q ), moving close to z axis we obtain the experimental data also presented.
1 1
3
0.35
Let perform the phase volume of the final state
Ourmodel
0.30 BaBar-2009 dΓ3 = (2π)−5δ4(p++p−−q+−q−−qπ)×
V CLEO-1998 × d3q+d3q−q3qπ, (16)
Ge 0.25 2E+2E−2Eπ
), to the form
2
Q 0.20
( Λ1/2 s,q2,M2
V dΓ = (2π)−5πdΓ dq2 1 π , (17)
2 3 q1 1 2s
Q 0.15 (cid:0) (cid:1)
where Λ(a,b,c)=a2+b2+c2−2(ab+ac+bc) and
0.10
d3q d3q
dΓ = + −δ4(q −q −q ). (18)
q1 2E 2E 1 + −
0 5 10 15 20 25 30 35 Z + −
Q2, GeV2 Performing the integration on the invariant mass square
ofthe leptonpair (we use the approximations ≪s and
1
s <M2) the relevant cross section have a form (details
Fig. 3: A fit of our approach (Eq. (13)) with resultant fit- 1
ting parameters A = 0.49, B = 0.23 for π0 production and in Appendix A):
the comparison with the experimental data of BaBar [5] and
CLEO [17] facilities. σee¯→π0l¯l = πα4V(s)2 1− M2 3 ln s − 5 . (19)
6 s m2 3
(cid:18) (cid:19) (cid:20) l (cid:21)
e–(p ) π (q ) l –(q )
– π –
IV. CONCLUSION
FromourpointofviewtheQCDcorrectionsconnected
withthevertexofinteractionofthehighlyvirtualphoton
γ*(q) γ (q1) with quarks is essential and pion can be considered as a
pointparticle. SoatratherlargevaluesofQ2 the details
of pion wave function becomes irrelevant.
For heavys quarksM =400MeV enteringthe heavy
e +(p_ ) l +(q+ ) pseudoscalarmesonsη′ tshe effect ofSudakovformfactor
becomes more weak.
Fig. 4: The annihilation channel of process e+e− →π0l+l−. In literature presents the alternative explanations(see
theendofSectionII)oftheexperimentaldataBaBar[5].
OntheFigure3werepresentanumericalestimationfit
III. ANNIHILATION CHANNEL of the BaBar data. We obtain the qualitative logaritm-
logarithmical growth (see Eq. (A7)) of the transitional
Let us consider now the annihilation channel depicted form-factor (13). On the plot we put the best fitting of
on Fig. 4: BaBar data with two adjustable parameters A and B.
The similar phenomena can take place as well for the
e+(p )+e−(p )→γ∗(q)→ case of scalar mesons production.
+ −
→π0(q )l+(q )l−(q ), (14) We remind as well the possibility to measure the
π + − transition pion form factor in electro-proton scattering
ep→eπ p. The relevant cross section will be
where l =e,µ and p2 =0, q2 =m2, q2 =M2, s=q2 = 0
± ± l π
(p+ +p−)2, s1 = q12 = (q+ +q−)2. We put the matrix dσep→eπ0p αg g 2 V2(Q2)
element of this process in the form: = ρqq ρNN ×
dQ2 8π(Q2+M2) Q2
(cid:18) ρ (cid:19)
(4πα)2 Q2
M = JµJν(l)V (s)ǫ qαqβ, × F2(Q2)+ F2(Q2) J(Q2),(20)
q2q2 µναβ 1 1 4M2 2
1 (cid:20) p (cid:21)
Jµ = v¯(p )γ u(p ),
+ µ − where F , F – are Dirac and Pauli proton form fac-
1 2
Jν(l) = v¯ (q )γ u (q ), (15) tors. Here instead of virtual photon the virtual vector
µ + ν µ −
mesontakes place; g , g are the ρ mesoncouplings
ρqq ρNN
wherequantityV(s)describesconversionoftwooffmass withquarksandnucleonscorrespondingly. Inthis casea
shell photons to the neutral pion (pion transition form- problem with background (ep → e∆+ → eπ0p) must be
factor) is defined in (2). overcomed.
4
We consider Sudakov form factor for time-like trans- To perform the integration we use Sudakov
fer momentum. With ordinary particles in the loop we parametrizationof the loop momentum:
must take into account the imaginary part of relevant
amplitude. For quarks inside a loop imaginary part is k =αn1+βn2+k⊥, (A2)
absent.
Taking into account the non-leading terms in expres- with n1,2 are light-like 4 vectors n2i = 0, transversal to
sionofSudakovexponentresultsinmodificationofquark k⊥,nik⊥ =0,buildedfromthe4-vectorsqπ,q1 suchthat
mass and a general shift of normalization: 2n1n2 =Q2. In such a parameterization
F q2,p21,p22 → AF q2,p21,p22 , (21) d4k = Q2dαdβd2k⊥. (A3)
2
(cid:0) Q2αβ(cid:1) > BM(cid:0)Q2, (cid:1)
Expressing the denominators of quark Green functions:
withA∼B ∼1canbeconsideredasafittingparameters
A,B >0. (k)=Q2αβ−~k2−M2+i0; (1)≈Q2α; (2)≈Q2β,
q
and performing the integration over k2 = −~k2 as (we
Acknowledgments ⊥
imply that principal part does no contribute)
We thank Dr. S. V. Mikhailov for his interest to the πd~k2
subject. =−iπ2 d~k2δ(−~k2+Q2αβ−M2)=
(k) q
Z Z
=−iπ2θ[Q2αβ−M2], (A4)
q
Appendix A: Details of calculation
we obtain
Transformation of the phase volume of 2-gamma cre- 1 1
ation process 2→3 consist in introduction of two trans- F Q2 = 1 dα dβ θ Q2αβ−M2 ×
ferred vectors q ,q: M2 2 α β q
1 (cid:18) q (cid:19) Mq2Z/Q2 Mq2Z/Q2 (cid:0) (cid:1)
dΓ3 = (2π)−5d4q1d4qd4qπd4p′+d4p′−× αsCF 1 1
× exp − ln ln . (A5)
× δ4 p −q −p′ δ4 p −q−p′ × 2π α β
− 1 − + + (cid:18) (cid:19)
× δ4((cid:0)q+q1−qπ)δ(cid:1) (q1(cid:0)−p−)2−m(cid:1)2e × Performing one integration we obtain:
(cid:16) (cid:17)
× δ (q−p+)2−m2e δ (q1+q)2−M2 , F(Q2 )= π Φ(z), (A6)
(cid:16) (cid:17) (cid:16) (cid:17) Mq2 αsCF
andusingtheSudakovparametrizationforthescattering
channel (6) we put it in form: where
s s 1
dΓ3 = (2π)−52dα1dβ1d2~q12dαdβd2~q× Φ(z)= dx 1−e−zx(1−x) ,
× δ sαβ−~q2−sβ−m2eα × Z0 x (cid:16) (cid:17)
× δ(cid:0)sα1β1−~q12−sα1−m(cid:1)2eβ1 × z = αsCFL2, L=ln Q2 .
× δ(cid:0)sαβ −(~q+~q )2−M2 .(cid:1) (A1) 2π Mq2
1 1
(cid:16) (cid:17) For large values of Q2 we obtain
Performing the integrations over α , α, β, we obtain the
1
result given above (see Eq. (7)). Q2 π
Expressionforscalarloopintegralwith3denominators F( )= (ln(z)+c), (A7)
M2 α C
and Sudakov vertex inserted has a form: q s F
Q2 d4kQ2V (Q2,p2,p2) where
F = − S 1 2 ,
(cid:18)Mq2(cid:19) Z iπ2 (k)(1)(2) 1 dx ∞ dx
(k) = k2−M2+i0; c= (1−e−x)− ≈0.57. (A8)
q x xex
(1) = (k+q )2−M2+i0; Z0 Z1
1 q
(2) = (k+q )2−M2+i0. In reality the quantity L ∼ 5 for light consistent quarks
π q
u,d which are present in the neutral pion and L∼1−2
where V (Q2,p2,p2) was defined in (4). for s-quark in η′ meson.
S 1 2
5
When considering the integration on the pair phase The differential cross section is
volume in annihilation channel we use the relation (con-
sequence of gauge invariance):
dσee¯→π0l¯l = α4Mq4 Λ3/2 s,q2,M2 dx ×
dΓ J(µ(l))(J(ν(l)))∗ = 24π3F2s5 1 x
q1 π
Xpol Z 1 (cid:0) 1 (cid:1)
× 1+ 1− , (A10)
=−13 gµν − q1µq2q1ν q12+2m2l πβ2−, (A9) (cid:18) 2x(cid:19)r x
(cid:18) 1 (cid:19)
(cid:0) (cid:1)
4m2
β = 1− l . where x=q2/ 4m2 >1.
− s q12 1 l
(cid:0) (cid:1)
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