Higher-twist contributions to neutrino-production of pions

0.22 MB·

by  B. Z. Kopeliovich

USM-TH-318
Higher-twist contributions to neutrino-production of pions
B. Z. Kopeliovich, Iv´an Schmidt and M. Siddikov
Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa,
y Centro Cient´ıfico - Tecnolo´gico de Valpara´ıso, Avda. Espan˜a 1680, Valpara´ıso, Chile
Inthispaperweestimatethesizeoftwist-3correctionstothedeeplyvirtualmesonproductionin
neutrinointeractionsduetothechiraloddtransversityGeneralizedPartonDistribution(GPD).We
concludethat incontrast topion electroproduction, inneutrino-inducedreactionsthesecorrections
are small. This happens due to large contribution of unpolarized GPDs H,E to the leading-
twist amplitude in neutrinoproduction. We provide a computational code, which can be used for
evaluation of the cross-sections accounting for these twist-3 corrections with various GPD models.
Our results are particularly relevant for analyses of the pion and kaon production in the Minerva
experiment at FERMILAB.
4
1 PACSnumbers: 13.15.+g,13.85.-t
0 Keywords: Singlepionproduction, generalizedpartondistributions,neutrino-hadroninteractions
2

n
I. INTRODUCTION
a
J

8 Nowadays one of the key objects used to parametrize nonperturbative structure of the target are the generalized
parton distributions (GPDs). For the kinematics where the collinear factorization is applicable [1, 2], they allow

]
evaluationofthecross-sectionsforawideclassofprocesses. TodayallinformationonGPDscomesfromtheelectron-
h
p protonandpositron-protonmeasurementsperformedatJLABandHERA,inparticularfromdeeplyvirtualCompton
- scattering(DVCS)anddeeplyvirtualmesonproduction(DVMP)[1–16]. TheforthcomingCLAS12upgradeatJLAB
p
will help to improve our understanding of the GPDs [16]. However,in practice, the procedure of extraction of GPDs
e
from experimental data is subject to the uncertainties, like large BFKL-type logarithms in the next-to-leading order
h
[ (NLO) corrections[17] in the HERA kinematics; contributions of the higher-twist components of GPDs and the pion
distribution amplitudes (DAs) in the JLAB kinematics [18–21], or uncertainties in vector meson DAs in case of ρ-
1
and φ-meson production.
v
From this point of view, consistency checks of the GPD extractionprocedure fromexperimental data, especially of
7
4 their flavorstructure, are important. Earlierwe proposedto study the GPDs in deeply virtualneutrinoproductionof
5 thepseudo-Goldstonemesons(π, K, η)[22]withthehigh-intensityNuMIbeamatFermilab,whichrecentlyswitched
1 to the so-called middle-energy (ME) regime with an average neutrino energy of about 6 GeV, and potentially is
. able to reach energies up to 20 GeV, without essential loss of luminosity. The νDVMP measurements with neutrino
1
0 and antineutrino beams are complementary to the electromagnetic DVMP. In the axial channel, due to the chiral
4 symmetry breaking we have an octet of pseudo-Goldstone bosons, which act as a naturalprobe of the flavor content.
1 Due to the V A structure of the chargedcurrent,in νDVMP one can access simultaneously the unpolarizedGPDs,
: H, E, and the−helicity flip GPDs, H˜ and E˜. Besides, using chiral symmetry and assuming closeness of parameters of
v
i pion and kaon, full flavour structure of the GPDs can be extracted.
X
It is worth reminding that the cross-sections were evaluated in [22] in the leading twist approximation, and for a
r correct extraction of the GPDs at the energies of MINERvA in ME regime, an estimate of the higher twist effects
a
is required. The first twist-3 correction arises due to contribution of the transversely polarized intermediate virtual
bosons and is controlled by convolution of poorly known transversity GPDs H , E , H˜ , E˜ and twist-3 DAs of
T T T T
pion. WhilethiscorrectionvanishesatasymptoticallylargeQ2,atmoderateQ2 inelectroproductionitgivesasizable
contribution, which was confirmed in the CLAS experiment [16]. In case of neutrino-productionsituation is different
since due to V A structure of the weak currents there is an additional and numerically dominant contribution of
−
the unpolarized GPDs H, E to the leading twist amplitude. In this paper we analyze the relative size of the twist-3
contributions to the neutrino-production of pions and demonstrate that they are indeed small. In this respect we
differ from [23], where the contribution of chiral odd GPDs was assumed to be numerically dominant.
The paper is organized as follows. In Section II we evaluate the Goldstone meson production by neutrinos on
nucleon targets accounting for higher twist effects. In Section III, for the sake of completeness we highlight the
properties of the GPD parametrization used for evaluations. In Section IV we present numerical results and make
conclusions.
2
II. CROSS-SECTION OF THE νDVMP PROCESS
The cross-sectionof the Goldstone mesons production in neutrino-hadron collisions has the form
dσ dσ dσ dσ
L T LT
=ǫ + + ǫ(1+ǫ)cosφ (1)
dtdx dQ2dφ dtdx dQ2dφ dtdx dQ2dφ dtdx dQ2dφ
B B B B
dσTT p dσL′T dσT′T
+ǫcos2φ + ǫ(1+ǫ)sinφ +ǫsin2φ ,
dtdx dQ2dφ dtdx dQ2dφ dtdx dQ2dφ
B B B
p
where t = (p p )2 is the momentum transfer to baryon, Q2 = q2 is the virtuality of the charged boson, x =
2 1 B
Q2/(2p q)isB−jorkenx, φistheanglebetweentheleptonandmeso−nproductionscatteringplanes,andweintroduced
·
shorthand notations
1 y γ2y2 2m x Q2
ǫ= − − 4 , γ = N B, y = .
1 y+ y2 + γ2y2 Q sxB
− 2 4
In the asymptotic Bjorken limit the cross-section is dominated by the first angular independent term
ǫdσ /dtdx dQ2dφ which was studied in our previous paper [22] and is a straightforward extension of the elec-
L B
troproductionofpionsstudiedin[20,21,24–28]. Aswewillseebelowthetwist-3correctionsaresmall,forthisreason
it is convenient to normalize all the cross-sections in (1) to this term,
dσ dσ
L
=ǫ (c cosnφ+s sinnφ) (2)
dtdx dQ2dφ dtdx dQ2dφ n n
B B
n
X
and discuss higher-twist effects in terms of harmonics c , s . In what follows, it is convenient to introduce a photon
n n
helicity matrix σ defined as
αβ
1
σαβ = 2 A∗ν′0,ναAν′0,νβ, (3)
νν′
X
where ν′0,να is the amplitude of the corresponding process in helicity basis, and ν, ν′ are the polarizations of the
A
initial and final baryon. In terms of σ the cross-sections in (1) can be written as
αβ
dσ
L
=Γσ (4)
dtdx dQ2dφ 00
B
dσ σ +σ σ σ
T =Γ ++ −− µ 1 ǫ2 ++− −− (5)
dtdx dQ2dφ 2 − − 2
B (cid:18) (cid:19)
p
dσ 1 ǫ
LT
=Γ Re(σ σ ) µ − Re(σ +σ ) (6)
dtdxBdQ2dφ 0+− 0− − r1+ǫ 0+ 0− !
dσ
TT
= ΓRe(σ ) (7)
dtdx dQ2dφ − +−
B
dσL′T 1 ǫ
= Γ Im(σ +σ )+µ − Im(σ σ ) (8)
dtdxBdQ2dφ − +0 −0 r1+ǫ −0− +0 !
dσT′T
= ΓIm(σ ) (9)
dtdx dQ2dφ − +−
B
where we introduced shorthand notations Γ and µ, which for the charged current (CC) and neutral current (NC)
are defined as
G2f2 x2 1 y+ y2 + γ2y2
F M B − 2 4
Γ = , (10)
CC 64π4Q2(1+(cid:16)Q2/M2 )2(1+γ2)(cid:17)3/2
W
G2f2 x2 1 y+ y2 + γ2y2
F M B − 2 4
Γ = , (11)
NC 64π4cos4θ Q2(cid:16)(1+Q2/M2)2(1+(cid:17)γ2)3/2
W Z
1
µ = , (12)
ν,ν¯
±2
3
f is the pion decay constant, G is the Fermi constant, θ is the Weinberg angle, and M , M are the masses of
π F W W Z
the W and Z bosons.
From (4-9) we can see that the terms sinφ and cosφ appear due to interference of the leading twist and
∼ ∼
twist-3 contributions, whereas all the other contributions appear entirely due to the twist-3 effects. Since the twist-3
amplitudes are suppressed by 1/Q compared to the leading twist result, we expect that in the large-Q limit
c , s 1/Q, c 1,c ,s 1/Q2. (13)
1 1 0 2 2
∼ − ∼
Due tothe factorizationtheoremthe amplitude ν′0,νβ in(3)maybe writtenasa convolutionofthe hardandsoft
A
parts,
+1
Aν′0,να =ˆ dx Hνq′′qλ′,νλCλq′′q0,λα, (14)
−1 q,q′=u,d,sλλ′
X X
where x is the averagelight-cone fraction of the parton, λ, q (λ′, q′) are the correspondinghelicity and flavour of the
initial (final) partons, the helicity amplitude, q′q is the process- and baryon-dependent soft part which will be
Hν′λ′,νλ
specified later, and q is the hard coefficient function.
Cλ′ν′,λν
In the leading twist, four GPDs, Hq′q, Eq′q, H˜q′q and E˜q′q contribute to q′q . They are defined as
Hν′λ′,νλ
P2¯π+ ˆ dzeixP¯+z B(p2) ψ¯q′ −2z γ+ψq 2z A(p1) = Hq′q(x,ξ,t)N¯(p2)γ+N(p1) (15)
D (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12) E (cid:16)
(cid:12)(cid:12) (cid:12)(cid:12) + ∆k Eq′q(x,ξ,t)N¯(p )iσ N(p )
2 +k 1
2m
N (cid:19)
P2¯π+ ˆ dzeixP¯+z B(p2) ψ¯q′ −z2 γ+γ5ψq 2z A(p1) = H˜q′q(x,ξ,t)N¯(p2)γ+γ5N(p1) (16)
D (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12) E (cid:16)
(cid:12)(cid:12) (cid:12)(cid:12) + ∆+ E˜q′q(x,ξ,t)N¯(p )N(p ) ,
2 1
2m
N (cid:19)
where P¯ = p +p , ∆ = p p and ξ = ∆+/2P¯+ x /(2 x ) (see e.g. [11] for the details of kinematics).
1 2 2 1 Bj Bj
− − ≈ −
In the case when the baryon remains intact, A = B, the corresponding GPDs are diagonal in the flavour space,
Hq′q δq′qHq, etc. In the generalcase,when A=B, in the right-handside (r.h.s.) of Eqs.(15), (16) there might be
∼ 6
extrastructureswhichotherwise areforbiddenbyT-parityin the caseofA=B [11]. Inwhatfollowswe assumethat
the target A is either a proton or a neutron, and the recoil B belongs to the same lowest SU(3) octet of baryons. In
this case, allsuch terms areparametricallysuppressed by the current quarkmass m and vanish in the limit of exact
q
SU(3),sowewilldisregardthem. Inthisspecialcase,wecanrelyontheSU(3)relationsandexpressthenondiagonal
transitional GPDs as linear combinations of the GPDs of the proton Hq, Eq, H˜q, E˜q [29], so (14) may be effectively
rewritten as
+1
Aν′0,να =ˆ dx Hνq′λ′,νλCλq′0,λα, (17)
−1 q λλ′
XX
The twist-3 correctionis controlled by the chiral odd transversity GPDs defined as [30]
P2¯π+ ˆ dzeixP¯+z B(p2) ψ¯q′ −2z iσ+jψq z2 A(p1) = (18)
= Hq′q(x,ξ,t)N¯D(p )iσ+j(cid:12)(cid:12)(cid:12)N((cid:16)p )+(cid:17)H˜q′qP¯+(cid:16)∆j(cid:17)−(cid:12)(cid:12)(cid:12)∆+P¯jE+Eq′qγ+∆j −∆+γj +E˜q′qγ+P¯j −P¯+γj ,
T 2 1 T m2 T 2m T m
(cid:18) (cid:19)
where j =1,2 is the transverse index. Similar to the leading-twist case, the flavour structure may be simplified with
the help of SU(3) relations and rewritten in terms of the transversity GPDs of the proton.
The matrix q in (17) is a linear combination of the helicity-odd and even GPDs,
Hν′λ′,νλ
Hνq′λ′,νλ = 21δλ−λ′ξ2 − (cid:0)1−−ξ(∆2(cid:1)1−H2im∆q2−)Eξq2Eq 1−(ξ∆21+2Him∆q2)−Eqξ2Eq !ν′ν (19)
p 1 ξ2 H˜q+ξ2E˜q ((cid:0)∆1+i∆2(cid:1))ξE˜q
+ sgn(λ) − − 2m +
(∆1−i∆2)ξE˜q 1 ξ2 H˜q ξ2E˜q ! !
(cid:0) (cid:1)2m − − ν′ν
+(mqν′νδλ,−δλ′,++nqν′νδλ,+δλ′,−),(cid:0) (cid:1)
4
where the coefficients mq and nq are given by
±,± ±,±
√ t′
mq = − 2H˜q +(1+ξ)Eq (1+ξ)E˜q , (20)
−− 4m T T − T
h t′ i
mq = 1 ξ2 H˜q, (21)
−+ − 4m2 T
p ξ2 ξ t′
mq = 1 ξ2 Hq Eq + E˜q H˜q , (22)
+− − T − 1 ξ2 T 1 ξ2 T − 4m2 T
(cid:20) − − (cid:21)
p
√ t′
mq = − 2H˜q +(1 ξ)Eq +(1 ξ)E˜q , (23)
++ 4m T − T − T
h i
√ t′
nq = − 2H˜q +(1 ξ)Eq +(1 ξ)E˜q , (24)
−− − 4m T − T − T
(cid:16) ξ2 ξ (cid:17) t′
nq = 1 ξ2 Hq Eq + E˜q H˜q , (25)
−+ − T − 1 ξ2 T 1 ξ2 T − 4m2 T
(cid:18) − − (cid:19)
p t′
nq = 1 ξ2 H˜q, (26)
+− − 4m2 T
p√ t′
nq = − 2H˜q +(1+ξ)Eq (1+ξ)E˜q , (27)
++ − 4m T T − T
(cid:16) (cid:17)
and we introduced a shorthand notation t′ = ∆2/(1 ξ2), where ∆ = p p is the transverse part of the
− ⊥ − ⊥ 2,⊥− 1,⊥
momentum transfer.
Evaluationofthe hardcoefficientfunction q isquite straightforward,andinthe leadingorderoverα isgiven
Cλ′0,λµ s
by the four diagrams shown schematically in Figure 1.
Figure 1: Leading-order contributions tothe DVMPhard coefficient functions.
A key ingredient in evaluation of the coefficient functions are the distribution amplitudes (DAs) of the produced
pion. Since we are interested in making evaluations up to twist-3 effects, we have to take into account both twist-2
and twist-3 pion DAs, defined respectively as [31]
1 du u u
φ (z) = ei(z−0.5)u 0 ψ¯ n nˆγ ψ n π(q) , (28)
2 if √2ˆ 2π −2 5 2
π
D (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12) E
(cid:12) (cid:12)
(cid:12) (cid:12)
1 m +m du u u
φ(p)(z) = u d ei(z−0.5)u 0 ψ¯ n γ ψ n π(q) , (29)
3 f √2 m2 ˆ 2π −2 5 2
π π
D (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12) E
(cid:12) (cid:12)
(cid:12) (cid:12)
3i m +m du u u
φ(σ)(z) = u d ei(z−0.5)u 0 ψ¯ n σ γ ψ n π(q) . (30)
3 √2f m2 ˆ 2π −2 +− 5 2
π π
D (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12) E
(cid:12) (cid:12)
As one can see from (28-30), the pseudovector and pseudoscala(cid:12)r DAs φ , φ(p) are chir(cid:12)al even (symmetric w.r.t.
2;π 3;π
z 1 z), whereas the tensor DA φ(σ) is chiral odd. Straightforwardevaluation of the diagrams shown in Figure 1
→ − 3;π
yields for the coefficient function
Cλq′0,λµ =δµ0δλλ′ ηAq,kc(k2)(x,ξ)+sgn(λ)ηVq,kc(k2)(x,ξ) + (31)
kX=±h i
m2
+δµ,+δλ,−δλ′,+(SAq −SVq)+δµ,−δλ,+δλ′,−(SAq +SVq)+O Q2 , (32)
(cid:18) (cid:19)
5
where we introduced shorthand notations
Sq = dz ηq c(3,p)(x,ξ) ηq c(3,p)(x,ξ) +2 ηq c(3,σ)(x,ξ)+ηq c(3,σ)(x,ξ) , (33)
A ˆ A+ + − A− − A− − A+ +
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
Sq = dz ηq c(3,p)(x,ξ)+ηq c(3,p)(x,ξ) +2 ηq c(3,σ)(x,ξ) ηq c(3,σ)(x,ξ) , (34)
V ˆ V+ + V− − V+ + − V− −
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
φ (z) 8πiα f 1
c(2)(x,ξ) = dz 2 s π , (35)
± ˆ z 9 Q x ξ i0
(cid:18) (cid:19) ± ∓
4πiα f ξ 1 φ (z) 4πiα f ξ 1 φ (z)
c(3,i)(x,ξ)= s π dz 3,i , c(3,i)(x,ξ)= s π dz 3,i ; (36)
+ 9Q2 ˆ z(x+ξ)2 − 9Q2 ˆ (1 z)(x ξ)2
0 0 − −
andtheprocess-dependentflavorfactorsηq , ηq arepresentedintableI.Aswasdiscussedabove,forthe processes,
V± A±
inwhicheitherinitialorfinalbaryonisdifferentfromproton,weusedSU(3)relations[29],validuptothecorrections
in current quark mass (m ). In the leading twist, due to the underlying SU(3) relations there are identities
q
∼ O
relating the neutrino-hadron and antineutrino-hadron cross-sections. In the next-to-leading order these relations
are broken due to the weak isospin-dependent factor µ in (12), and corresponding SU(3) identities are valid only
for the cross-sections dσTT and dσT′T (harmonics c2, s2). For certain processes either η+q or η−q might vanish. In
this case from the definition (33,34) we may see that S = S , and the matrix element σ which controls the
A V +−
±
cross-sections dσTT, dσT′T (7,9) vanishes identically.
Using symmetry of φ and antisymmetry of φ with respect to charge conjugation, we can show that dependence
p σ
onthepionDAsfactorizesinthecollinearapproximationandcontributesonlyastheminusfirstmomentofthelinear
combination of the twist-3 DAs, φ (z)+2φ (z),
p σ
1 φ(p)(z)+2φ(σ)(z)
φ−1 = dz 3 3 . (37)
3 ˆ z
0
(cid:10) (cid:11)
We see from (17, 36) that, excluding the very special case when all the transversity GPDs vanish at x = ξ, the
±
transverse amplitude suffers from a collinear singularity at these two points. In order to regularize it, we follow [19]
and introduce a small transverse momentum of the quarks inside the meson. Such regularizationmodifies (36) to
4πiα f ξ 1 φ (z, l )
c(3,i)(x,ξ)= s π dzd2l 3,i ⊥ , (38)
+ 9Q2 ˆ0 ⊥(x+ξ i0) z(x+ξ)+ 2ξl2⊥
− Q2
4πiα f ξ 1 (cid:16)φ (z, l ) (cid:17)
c(3,i)(x,ξ)= s π dzd2l 3,i ⊥ , (39)
− 9Q2 ˆ0 ⊥(x ξ+i0) (1 z)(x ξ) 2ξl2⊥
− − − − Q2
(cid:16) (cid:17)
where l is the transverse momentum of the quark, and we tacitly assume absence of any other transverse momenta
⊥
in the coefficient function.
III. GPD AND DA PARAMETRIZATIONS
The pion DAs are one of the main sources of uncertainty in the present analysis. For the leading twist DA φ (x),
2π
thecurrentlyavailabledataonmesonphotoproductionformfactorF Q2 arecompatiblewithanasymptoticform
πγγ
φ (z)=6√2f z(1 z), with a typical uncertainty in the minus-first moment of the orderof 10% (see e.g. [32, 33]
as π − (cid:0) (cid:1) ∼
and reviews in [34, 35]).
For the twist-3 contribution, as was discussed in Section II, the DAs φ (z,l ) and φ (z,l ) contribute in a
3;p ⊥ 3;σ ⊥
linear combination
φ (z, l )=φ (z, l )+2φ (z, l ). (40)
3 ⊥ 3;p ⊥ 3;σ ⊥
For the sake of simplicity, we parametrize (40) in the form
2a3
φ (z, l )= p l φ (z)exp a2l2 , (41)
3 ⊥ π3/2 ⊥ as − p ⊥
(cid:0) (cid:1)
6
Table I: The flavour coefficients ηq for various processes (q =u,d,s,...). For the case of CC mediated processes, take ηq =
V
ηq, ηAq =−ηq. Forthecaseof±NCmediatedprocesses,takegq correspondingtovectorcurrentcouplinggVq andaxial-ve±ctor
cu±rrent c±oupling±gq for thehelicity odd and even GPDs respectively.
A
Process type ηq ηq Process type ηq ηq
+ +
− −
νp→µ−π+p CC Vudδqu Vudδqd νn→µ−π+n CC Vudδqd Vudδqu
ν¯p→µ+π−p CC Vudδqd Vudδqu ν¯n→µ+π−n CC Vudδqu Vudδqd
ν¯p→µ+π0n CC Vudδqu√−2δqd −Vudδqu√−2δqd νn→µ−π0p CC −Vudδqu√−2δqd Vudδqu√−2δqd
νp→νπ+n NC gd(δqu−δqd) gu(δqu−δqd) νn→νπ−p NC gu(δqu−δqd) gd(δqu−δqd)
νp→νπ0p NC guδqu−gdδqd guδqu−gdδqd νn→νπ0n NC guδqd−gdδqu guδqd−gdδqu
√2 √2 √2 √2
ν¯p→µ+π−Σ+ CC −Vus(δqd−δqs) 0 ν¯n→µ+π−Λ CC −Vus2δqd−√δq6u−δqs 0
ν¯p→µ+π0Σ0 CC Vu2s (δqd−δqs) 0 ν¯n→µ+π−Σ0 CC −Vusδqu√−2δqs 0
ν¯p→µ+π0Λ CC Vus2δqu−2δ√q3d−δqs 0 ν¯n→µ+π0Σ− CC Vusδqu√−2δqs 0
νp→µ−K+p CC Vusδqu Vusδqs νn→µ−K+n CC Vusδqd Vusδqs
ν¯p→µ+K−p CC Vusδqs Vusδqu ν¯n→µ+K−n CC Vusδqs Vusδqd
ν¯p→µ+K0Σ0 CC 0 −Vudδqd√−2δqs ν¯n→µ+K0Σ− CC 0 −Vud(δqu−δqs)
ν¯p→µ+K0Λ CC 0 −Vud2δqu−√δq6d−δqs νn→νK0Λ NC −gd2δqd−√δq6u−δqs −gd2δqd−√δq6u−δqs
ν¯p→µ+K¯0n CC 0 −Vus(δqu−δqd) νn→νK0Σ0 NC −gdδqu√−2δqs −gdδqu√−2δqs
νp→µ−K+Σ+ CC 0 −Vud(δqd−δqs) νn→µ−K+Σ0 CC 0 −Vudδqu√−2δqs
νp→νK+Λ NC −gd2δqu−√δq6d−δqs −gu2δqu−√δq6d−δqs νn→µ−K+Λ CC 0 −Vud2δqd−√δq6u−δqs
νp→νK+Σ0 NC gdδqd√−2δqs guδqd√−2δqs νn→µ−K0p CC Vus(δqu−δqd) 0
νp→νK0Σ+ NC −gd(δqd−δqs) −gd(δqd−δqs) νn→νK+Σ− NC −gd(δqu−δqs) −gu(δqu−δqs)
νp→νηp NC guδqu+gdδqd−2gdδqs guδqu+gdδqd−2gdδqs νn→νηn NC guδqd+gdδqu−2gdδqs guδqd+gdδqu−2gdδqs
√6 √6 √6 √6
ν¯p→µ+ηn CC Vudδqu√−6δqd Vudδqu√−6δqd ν¯n→µ+ηΣ− CC −Vusδqu√−6δqs 2Vusδqu√−6δqs
ν¯p→µ+ηΣ0 CC Vusδqu2√−3δqd −Vusδqu√−3δqd νn→µ−ηp CC Vudδqu√−6δqd Vudδqu√−6δqd
ν¯p→µ+ηΛ CC Vus2δqu−δ6qd−δqs −Vus2δqu−δ3qd−δqs
where the numerical constant a is taken as a 2GeV−1 in analogy with [19, 20].
p p
≈
More than a dozen of different parametrizations of GPDs have been proposed in the literature [7, 12, 28, 36–44].
Whileweneitherendorsenorrefuteanyofthem,forthesakeofconcretenessweusetheparametrization[26–28],which
successfully described HERA [45] and JLAB [26–28] data on electroproduction of different mesons, so is expected to
provide a reasonable description of νDVMP. The parametrization is based on the Radyushkin’s double distribution
ansatz. It assumes additivity of the valence and sea parts of the GPDs,
H(x,ξ,t)=H (x,ξ,t)+H (x,ξ,t),
val sea
which are defined as
3θ(β) (1 β )2 α2
Hq = dβdαδ(β x+αξ) −| | − q (β)e(bi−αiln|β|)t, (42)
val ˆ − 4(1 β )3 val
|α|+|β|≤1 (cid:0) −| | (cid:1)
3sgn(β) (1 β )2 α2 2
Hq = dβdαδ(β x+αξ) −| | − q (β)e(bi−αiln|β|)t, (43)
sea ˆ − 8(1 β )5 sea
|α|+|β|≤1 (cid:0) −| | (cid:1)
and q and q are the ordinary valence and sea components of PDFs. The coefficients b , α , as well as the
val sea i i
parametrization of the input PDFs q(x), ∆q(x) and pseudo-PDFs e(x), e˜(x) (which correspond to the forward limit
of the GPDs E, E˜) are discussed in [26–28]. The unpolarized PDFs q(x) are adjusted to reproduce the CTEQ PDFs
inthelimitedrange4.Q2 .40GeV2. Noticethatinthismodeltheseaisflavorsymmetricforasymptoticallylarge
7
Q2,
Hu =Hd =κ Q2 Hs , (44)
sea sea sea
where (cid:0) (cid:1)
0.68
κ Q2 =1+ , Q2 =4GeV2.
1+0.52ln(Q2/Q2) 0
0
(cid:0) (cid:1)
The equalityofthe seacomponentsofthe lightquarksin(44)shouldbe consideredonlyasaroughapproximation,
since in the forward limit the inequality d¯ = u¯ was firmly established by the E866/NuSea experiment [46]. For
6
this reason, the predictions made with this parametrization of the GPDs for the p ⇄ n transitions in the region
x (0.1...0.3) might slightly underestimate the data.
Bj
∈
The transversity GPDs in the parametrization [20] are obtained using a familiar double-distribution based
parametrization (42,43), and the forwardlimit of these GPDs is parametrized as
Ha(x,0,0)=Na √x(1 x)(qa(x)+∆qa(x)),
T HT −
E¯ (x,0,0) E (x,0,0)+2H˜ (x,0,0)=Na x−α(1 x)β,
T ≡ T T E¯T −
where the values of the parameters Na,α,β are fixed from the lattice data. Since in the parametrization[20], as well
i
as in any other parametrization of chiral-odd GPDs available in the literature, s-quarks are not included, we do not
make any predictions for strangeness production.
IV. NUMERICAL RESULTS AND DISCUSSION
In this section we would like to present numerical results for the twist-3 corrections to pion production using
the Kroll-Goloskokov parametrization of GPDs [19, 26–28], briefly discussed in the section III. As was discussed
in section II, one of the consequences of the higher twist corrections is the appearance of the azimuthal angular
dependence of the DVMP cross-sections Since the twist-3 corrections are small, in what follows we prefer to discuss
the results in terms of the angular harmonics c , s defined in (2). The most important harmonics is c , because its
n n 0
deviationfromunity affects the extractionofGPDs in the leading twist approximation,andextractionof the leading
twist result requires the experimentally challenging Rosenbluth separation with varying energy neutrino beam. All
the other harmonics generate nontrivial angular dependence and can be easily separated from the leading-twist
contribution. For example, the angle-integrated cross-sectiondσ/dlnx dtdQ2 is not sensitive to those harmonics at
B
all.
In Figure 2 we show the harmonics c , s for processes without change of baryon state. The processes shown in
n n
the lowerrowareisospinconjugate to processesinthe upper row. While in the leadingtwist the cross-sectionsof the
former and the latter coincide, with the accountof the twist-3 correctionsthis is no longer valid due to the difference
in weak isospin of ν and ν¯. In all cases at x . 0.5, where the cross-section is the largest, the harmonics are small
B
and do not exceed few per cent. The largest twist-3 contribution is due to the c harmonics, which may reach up to
1
twentypercent. Thisisdifferentfromtheelectroproductionexperiments,wherec ( σ )isverysmall. Thisresult
1 LT
∼
can be understood from (6): due to parity nonconservation in weak interactions we have for the interference term
σ =σ . A positive value of c for mostprocessesimplies thatpion productioncorrelateswith the directionof the
0+ 0− 1
6
produced muon (scattered neutrino) in the case of CC (NC) mediated processes. The interference term also yields a
relatively large harmonics s which appears due to the interference of the vector and axial vector contributions.
1
In the regionof x&0.5all the harmonicsincrease,but the cross-sectionsfor both the leading twistand subleading
twist results are suppressed there due to increase of t and are hardly accessible with ongoing and forthcoming
min
| |
experiments.
In the Figure 3 we present the harmonics c , s for processes with change of the baryon state. As was discussed
n n
in [22], in the leading twist these processesaresensitive to the valence quarksdistributions. As we cansee,similar to
the previous case, all the harmonics are small and except the region of x 1 do not exceed few per cent.
B
∼
Forstrangenessproductionweobtainedqualitativelysimilarresults(correctionsaresmall),howeverwerefrainfrom
making predictions because the corresponding amplitudes are sensitive to the strange component of the chiral odd
GPDs which are unknown at this moment.
V. CONCLUSIONS
In this paper we estimated the contributions of the twist-3 corrections due to the chiral odd GPDs. One of
the manifestations of the twist-3 corrections is the appearance of the dependence on the angle between the lepton
8
y=0.5, Q2=4 GeV2, ∆ =0.2 GeV y=0.5, Q2=4 GeV2, ∆ =0.2 GeV y=0.5, Q2=4 GeV2, ∆ =0.2 GeV
t_r (a) tr (b) tr (c)
c-1 νp→µ+π-p c-1 νp→µ-π+p c-1 νp→νπ0p
100 c-c01 100 |-cc01| 100 cc01
s2 -s2 0 |s2|
s12 s21 c<1 s21
10-1 10-1 c1>0 10-1
sn sn sn
c , n 10-2 c , n 10-2 c , n 10-2 s1<0 s1>0
10-3 10-3 10-3
10-4 10-4 10-4
0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8 0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8 0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8
x x x
Bj Bj Bj
y=0.5, Q2=4 GeV2, ∆ =0.2 GeV y=0.5, Q2=4 GeV2, ∆ =0.2 GeV y=0.5, Q2=4 GeV2, ∆ =0.2 GeV
tr (d) t_r (e) tr (f)
c-1 νn→µ-π+n c-1 νn→µ+π-n c-1 νn→νπ0n
100 c-sc012 100 |--ccs012| <0 100 cc|s012|
s12 s21 c1>0 c1 s21
10-1 10-1 10-1
sn sn sn
c , n 10-2 c , n 10-2 c , n 10-2 s1<0 s1>0
10-3 10-3 10-3
10-4 10-4 10-4
0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8 0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8 0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8
x x x
Bj Bj Bj
Figure2: (coloronline)Pionproductiononnucleonswithoutchangeofthebaryonstate. Processesinthelowerrowdifferfrom
theprocesses in theupperrow dueto isospin conservation breakdown byhigher-twist corrections.
y=0.5, Q2=4 GeV2, ∆ =0.2 GeV y=0.5, Q2=4 GeV2, ∆ =0.2 GeV
tr (a) t_r (b)
c-1 νp→νπ+n c-1 νp→µ+π0n
100 |cc01| 100 cc01
|s2| |s2|
-s1 -s1
2 2
10-1 10-1
>0
s 1
sn sn s1<0
c , n 10-2 s>0 c , n 10-2
s<0 1
c<0 c>0 1
10-3 1 1 10-3
10-4 10-4
0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8 0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8
x x
Bj Bj
y=0.5, Q2=4 GeV2, ∆ =0.2 GeV y=0.5, Q2=4 GeV2, ∆ =0.2 GeV
tr (c) tr (d)
c-1 νn→νπ-p c-1 νn→µ-π0p
100 cc01 100 cc01
|s2| |s2|
-s1 -s1
2 2
10-1 10-1
c , snn 10-2 s1<0 s>10 c , snn 10-2 s1<0 s>10
10-3 10-3
10-4 10-4
0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8 0(c) .M. Siddikov1 0.2 0.3 0.4 0.6 0.8
x x
Bj Bj
Figure 3: (color online) Pion production on nucleons with change of the baryon state. Processes in the lower row differ from
theprocesses in theupperrow dueto isospin-breaking by higher-twist corrections.
9
scattering and pion production planes. We found that the largest harmonics is c , which can reach up to twenty
1
per cent, however it does not affect the angular integrated cross-section dσ/dx dtdQ2. All the other harmonics
B
are small and do not exceed few per cent. This happens because in case of neutrino interactions, in contrast to
electroproduction of pions, there are large contributions of unpolarized GPDs H, E to the leading-twist amplitude.
Notice that the similar angular harmonics may be generated by interference of the leading twist result with the
electromagnetic corrections [47]. At moderate virtualities of the order a few GeV2 this mechanism also gives small
harmonics (of the order few per cent), however those corrections grow rapidly as a function of Q2, and already at
Q2 100GeV2 electromagnetic mechanism becomes dominant.
∼
Tosummarize,weconcludethatdeeplyvirtualproductionofpionsandkaonsonprotonsandneutronsbyneutrinos
with typical values of Q2 of the order few GeV2 provide a clean probe for the GPDs, with various corrections of the
orderoffewpercent. OurresultsarerelevantforanalysisofthepionandkaonproductionintheMinervaexperiment
at FERMILAB as well as for the planned Muon Collider/Neutrino Factory [48–50]. An ideal target for study of the
GPDscouldbealiquidhydrogenordeuterium. Forothertargetsthereisanadditionaluncertaintyduetothenuclear
effects which will be addressed elsewhere.
Weprovideacomputationalcode,whichcanbeusedforevaluationofthecross-sectionswithinclusionofthetwist-3
corrections employing various GPD models.
Acknowledgments
This work was supported in part by Fondecyt (Chile) grants No. 1130543,1100287and 1120920.
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