Diffractive neutrino-production of pions on nuclei: Adler relation within the color-dipole description

0.21 MB·English

by  B. Z. Kopeliovich

USM-TH-296
Diffractive neutrino-production of pions on nuclei:
Adler relation within the color-dipole description
B.Z. Kopeliovich,∗ Iván Schmidt,† and M. Siddikov‡
Departamento de Física, Centro de Estudios Subatómicos, y Centro Científico - Tecnológico de Valparaíso,
Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
Effects of coherence in neutrino-production of pions off nuclei are studied employing the color
dipole representation and path integral technique. If the nucleus remains intact, the process is
controlled by the interplay of two length scales. One is related to the pion mass and is quite long
(at low Q2), while theother, associated with heavy axial-vector states, is much shorter. The Adler
relationisfoundtobebrokenatallenergies,butespecially stronglyatν &10GeV,wherethecross
section is suppressed by a factor ∼ A−1/3. On the contrary, in a process where the recoil nucleus
2 breaksup intofragments, theAdlerrelation turnsout tobestrongly brokenat low energies, where
1 thecross section is enhanced by a factor ∼A1/3, but has a reasonable accuracy at higher energies,
0 where all the coherence length scales become long.
2
PACSnumbers: 13.15.+g,13.85.-t
n
Keywords: Diffractiveneutrinointeractions, Adlerrelation, Single-pionproduction
a
J

9 I. INTRODUCTION previous calculations performed in [5], which contra-
1
dicted the AR and which were based on an incorrect

model for nuclear effects (see critical discussion in [4]).
] Due to the V A structure of weak interactions high-
h energy neutrinos−serve as a source of the axial current. Recently,alargedeviationfromtheARpredictionsfor
p Unfortunately, because of the smallness of the neutrino- coherent and incoherent diffractive neutrino-production
-
p hadroncross-sectionsexperimental data have been quite of pions at high energies was discovered in [6] using a
e scarce until recently, mostly being limited to the total simple two-channel toy model which contains only ax-
h cross-sections. Withthelaunchofthenewhigh-statistics ial meson and pion. This deviation is caused by initial
[
experiments like MINERνA at Fermilab [1], now the andfinalstateinteractions,calledabsorptivecorrections,
1 neutrino-hadroninteractions may be studied with a bet- which are very strong for large rapidity gap processes
v ter precision. like diffractive production. The onset of these correc-
3 The properties of the vector current have been well tions is controlled by the coherence length lc = 2ν/m2a,
5 studied, mostly in collisionsof chargedleptons with pro- where ν is the transferred energy (or the pion energy),
0
4 tons and nuclei in processes of deep inelastic scattering andma ∼1GeV. TheARisatworkonlyifthecoherence
1. (CDoImSp),todneespclayttveirritnugal(RCComS)patonndsmcaetstoenripnrgod(DucVtCioSn).,Trehael lie.en.gtaht lioswsheonretrgcoiems.paArtedhitgohetrheenneurcglieeasrthsiezev,alluce≪ofRthAe,
0 structure of the axial current is less known. cross section considerably drops, by a factor A−1/3,
2 ∼
1 According to Adler relation (AR) [2, 3], the cross sec- comparedto the ARprediction. In this paper we extend
: tion of neutrino interaction at zero virtuality is propor- theresultobtainedin[6]anddemonstratethatitisvalid
v tionaltothecross-sectionofpioninteractiononthesame in a realistic color dipole model.
i
X target and with the same final hadronic state, The nuclear shadowing effect for the total neutrino-
nucleus interaction at low Q2 was first calculated in [7]
r
a dσ G2 E ν within a specific optical model, which was essentially
νp→lF = F f2 ν − σ (ν), (1)
dνdQ2 (cid:12) 2π2 π E ν πp→F oversimplified. ItwasalsocalculatedwithintheGlauber-
(cid:12)Q2=0 ν Gribov theory [8, 9] in [10–12], and in this case it gave
(cid:12)
where G = 1(cid:12).166 10−5GeV−2 is the electro-weak good agreement with data from the WA59 experiment
F × [13]. This calculation was based on the AR, which in
Fermi coupling; F denotes the final hadronic state; E
ν this case has no absorptive corrections, and is expected
and ν are the energy of the neutrino and transferred en-
to be rather accurate.
ergy in the target rest frame, respectively.
In this paper we describe the neutrino-nucleus inter-
Nucleareffectsindiffractiveneutrino-productionofpi-
actions within the color dipole approach which was pro-
ons, coherent (νA πA) and incoherent (νA πA∗),
→ → posedin[14]fordescriptionofthehigh-energyscattering
were calculated in [4] based on the AR and Glauber
processes. The dipole representation is especially simple
eikonal approximation. The results were different from
and effective at high energies, where the dipole separa-
tion does not fluctuate during propagation through the
nucleus,being”frozen” byLorentztimedilation. Besides,
athighenergies,orsmallBjorkenxindeep-inelasticscat-
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] tering (DIS), gluonic exchanges with the target domi-
‡Electronicaddress: [email protected] nate in the scattering amplitude. The phenomenological
2
dipole cross section is usually fitted to HERA data for target. More details can be found in [6]. The pion pro-
the proton structure function at small x, and it is risky ductioncross-sectionintheneutrino-protoncollisionshas
to use at at lower energies, where Reggeons, i.e. quark- the form
antiquark exchanges become important, and should be
eaaxtnpdelniocenirteglyieshsaodaudsledldowrteofaestrh√teosdo.iptho2elerGcmerVoosdtsehlsesecmwthiooidnceh.l cEisovnnetonattiunvaaelllxiyd-, ddt3dσQνp2→dxlπBpj = 3G22Fπx3Qy24 1Lµν(cid:0)qW2 µA2→π(cid:1)1∗+Wν4Am→2xπ2Bj, (2)
(cid:16) − MW2 (cid:17) q Q2
plicit contributions of resonances (see e.g. [15, 16] and a
recentreviewin[17]). Weperformamorerigorouscalcu- where L is the axial lepton tensor, and WA→π is the
lationthanwasdonein[6],wherethedifferentcoherence µν ν
amplitudeofpionproductionbyaxialcurrentonthepro-
lengths were introduced by hand.
ton target. In the color dipole model this amplitude has
The color dipole approach was tested in a number
the form
of photon-nucleon and photon-nucleus processes, such
as Deep Inelastic Scattering [18, 19], Drell-Yan reaction
1
[20] heavy meson production [21], as well as deeply vir- WA→π s,∆,Q2 = g qµqν dβ dβ (3)
tual Compton scattering (DVCS), real Compton scat- µ (cid:18) µν − q2 m2(cid:19)ˆ 1 2
(cid:0) (cid:1) − π
tering (RCS), double deeply virtual Compton scattering 0
(DDVCS) on the nucleons and nuclei (See e.g. [22–28]), d2r d2r Ψ¯π(β ,~r )
giving a reasonable description of the total and differen- × ˆ 1 2 f 2 2
tialcross-sections. Also, the colordipole modelhas been d(β ,~r ;β ,~r ;∆)Ψi (β ,~r )
appliedto the descriptionofthe neutrino physics in [29– × A 1 1 2 2 ν 1 1
36]. Single-pion production by neutrinos on a proton where Ψ¯π and Ψi are the distribution amplitudes of the
target has been studied recently within the color dipole f ν
pionandaxialcurrentrespectively;∆isthe4-momentum
approach in [37].
transfer in the dipole-proton scattering, , and d(...) is
Here we extend the results obtained for neutrino- A
the dipole scattering amplitude.
proton interactions in [37] to the nuclei using the
Thedistributionamplitudes areessentiallynonpertur-
Glauber-Gribov approach [8, 9]. While in the high-
bativeobjects. Weparametrizethemintheformderived
energy (“frozen”) regime the shadowing corrections are
in[38–42]. Thedipolescatteringamplitude d(...)in(3)
given by the trivial exponential attenuation factor, we A
is a universal object, which depends only on the target,
use anapproachwhichis alsovalidatintermediateener-
but not on the projectile and final states. In addition
gies, where the dipole size fluctuates during propagation
to the axial current contribution, in (3) the contribu-
throughthe nucleus. As wasdiscussedearlier,we do not
tion of the vector current should be also present. This
consider the region of very low energies (√s . 2 GeV)
contribution involves a poorly known helicity flip dipole
due to limitations of the model and absence of explicit
amplitude ˜ , which is small [43] anyway, and therefore
s-channel resonances in the model. we neglectAitd. Moreover, at small Q2 the vector current
Thepaperisorganizedasfollows. InSectionIIforthe
contribution is suppressed by a factor that goes as Q2.
sakeofcompletenesswegivetheformulaswhichareused
for evaluation of the color dipole amplitudes on the pro- At high energies in the small angle approximation,
ton. In Section III we discuss the framework which was ∆/√s 1,thequarkseparationandfractionalmomenta
≪
used for evaluation of nuclear corrections. In Section IV β are preserved, so
we present results and draw conclusions.
d β ,~r ;β ,~r ;Q2,∆ δ(β β )δ(~r ~r ) (4)
1 1 2 2 1 2 1 2
A ≈ − −
(cid:0) (cid:1)
(ǫ+i) mfN ~r,∆~,β,s
II. DIFFRACTIVE PION PRODUCTION ON A × ℑ q¯q(cid:16) (cid:17)
PROTON
where ǫ is the ratio of the real to imaginary parts, and
Inthissectionwepresentabriefsurveyoftheformulas forthe imaginarypartofthe elastic dipole amplitude we
for evaluation of the neutrino cross-section on a proton employ the model developed in [22, 44–46],
ℑmfq¯Nq(cid:16)~r,∆~,β,s(cid:17)= σ04(s)exp(cid:20)−(cid:18)B2(s) + R102(6s)(cid:19)∆~2⊥(cid:21)(cid:18)e−iβ~r·∆~ +ei(1−β)~r·∆~ −2ei(21−β)~r·∆~e−R20r2(s)(cid:19). (5)
and the phenomenological functions σ (s), R2(s), B(s) arefittedtoDIS,realphotoproductionandπpscattering
0 0
3
data. or a ρπ pair. The latter has an invariant mass distribu-
Intheforwardlimit, ∆ 0,the imaginarypartofthe tion,whichpeaksclosetothea mass,andcanbetreated
1
→
amplitude (4) reduces to the saturated form [18] of the as an effective a-pole [4, 6, 50–52],
dipole cross section,
2ν
la = . (10)
σ (r,s) = mfN ~r,∆~,β,s (6) c Q2+m2
d ℑ q¯q(cid:16) (cid:17) a
r2 For large virtuality Q2 m2 the nuclear effects depend
= σ0(s)(cid:18)1−exp(cid:18)−R2(s)(cid:19)(cid:19). on one coherence lengt≫h lπa la. However, for m2 .
0 c ≈ c π
Q2 m2 we have lπ la, so there are three different
The calculation of the differential cross section also ≪ a c ≫ c
regimes for the coherence effects.
involves the real part of scattering amplitude, which ac-
cording to [47] is related to the imaginary part as For la lπ R the coherence length is small
• c ≪ c ≪ A
andthereisnoshadowing. The cross-sectionofco-
π ∂ mf(∆=0)
ef(∆=0)=sαtan α 1+ ℑ . herentpionproduction(thenucleusremainsintact)
R (cid:20)2 (cid:18) − ∂lns(cid:19)(cid:21) sα
vanishes, and the incoherent cross section (the nu-
(7)
cleus decays to fragments) is a simple sum of the
In the model under consideration the imaginary part of
cross-sections on separate nucleons.
the forwarddipole amplitude indeed has a power depen-
dence on energy, mf(∆ = 0; s) sα, so (7) simplifies In the intermediate regime of lπ R , la R
to I ∼ • theq¯qdipoleisproducedinstantcan≫eousAlyincsi≪dethAe
e nucleus and then evolves into the pion wave func-
R A =tan π(α 1) ǫ. (8) tion. For the distribution amplitude of the dipole,
m 2 − ≡
ℑ A (cid:0) (cid:1) wemayusethedistributionamplitudeevaluatedin
This fixes the phase of the forward scattering ampli- the IVM.
tude, which we retain for nonzero momentum transfers,
assumingsimilardependencesforthe realandimaginary If lπ la R the axial current fluctuates into
• c ≫ c ≫ A
parts. a q¯q dipole long before the production of the pion,
andthismesonmayscatteronthenucleons. Inthis
regime one can treat the dipole size as “frozen” by
III. NUCLEAR EFFECTS Lorentz time dilation, what considerably simplifies
the calculations. As was discussed in detail in [6],
A. Quark shadowing the Adler theorem (1) is severely broken in this
regime, even for Q2 = 0, due to large absorptive
corrections.
Nuclearshadowinginhardreactionsoriginatesmainly
fromthe contributionofsoftinteractions(if any). Inthe
The theoretical description of the transition region,
color dipole model, the soft contribution arises from the
wherethelifetimeofaq¯qfluctuationcannotbeeitherne-
so called aligned jet configurations [48], corresponding
glected or considered to be sufficiently long to apply the
to q¯q fluctuations very asymmetric in sharing the pho-
“frozen” sizeapproximation,isthe mostdifficult task. In
ton momentum, β 1. Such fluctuations, having large
≪ this regime a q¯q dipole propagates through the nuclear
transverse separation, are the source of quark shadow-
mediumwithavaryingsize. Inthispaperweemploythe
ing[49]. Theyaresuppressesforlongitudinallypolarized
description of shadowing developed in [53] and based on
currents,anddonotexistif the hardscaleis imposedby
the light-cone Green function technique [54]. The prop-
the heavyquarkmassratherthanbyvirtualityQ2. This
agation of a color dipole in the nuclear medium is de-
clearly shows that quark shadowing is a higher twist ef-
scribed as a motion in an absorptive potential,
fect. The leading twist shadowing effects arise from the
higher Fock components containing gluons, q¯qg . In- ∂ ∆ (r ) ε2
deed, the gluon carries a small fraction of the|totial mo- i + ⊥ 2 − +U(r2,z2) G(z2,~r2;z1,~r1)
(cid:20) ∂z 2να(1 α) (cid:21)
mentum,thereforethe meantransverseseparationofthe 2 −
q¯q and gluon is large even at high Q2. We provide more = iδ(z2 z1)δ(2)(~r2 ~r1). (11)
− −
details on gluon shadowing in the next section.
where the Green function G(z ,~r ;z ,~r ) describes the
As was discussed in [6], neutrino-production of pions 2 2 1 1
probabilityamplitude forthepropagationofdipolestate
onnucleiiscontrolledbytwocharacteristiclengthscales,
with size r at the light-cone starting point z to the
the coherence length for pion production, 1 1
dipole state with size r at the light-cone point z ; ǫ2 =
2 2
lcπ = Q22+νm2, (9) pαo(1te−nαti)aQl d2+esmcri2qb,easnadbtshoerpimtioangininartyhepanrutcolefatrhemleigdhiut-mco,ne
π
and the coherence length related to excitation in the in- 1
termediatestateofaxial-vectorstates,likethea1-meson, ℑmU(r,z)=−2σq¯q(r)ρA(b,z). (12)
4
In this paper we assume that the realpartofthe scat- Thefirstterminsidethesquarebracketsin(15)issup-
tering potential is zero. This approximation is justified pressedas (m),sincethetransitionfromspin-1tospin-
O
for large Q2, and for small Q2 the real part should be 0 state requires helicity flip for one of the quarks in the
added, as was done in [55]. quark-antiquarkpair,sotheamplitudemaybe rewritten
Thentheshadowingcorrectiontotheamplitudeofthe as
coherent pion production gets the form
(∆)= dzd2bρ (b,z)eib·∆(F (b,z) F (b,z)), 1
A ˆ A 1 − 2 (s,∆ ) 2 d2bei∆~⊥·~b⊥ dαd2rΨ¯ (α,r)Ψ (α,r)
(13) A ⊥ ≈ − ˆ ˆ f in
where 0
+∞
1
× exp−2σq¯q(r) ˆ dζρA(ζ,b) (16)
F1(b, z) = ˆ dαd2r1d2r2Ψ¯f(α,r2)G(+∞,~r2;z,~r1) AnothercasewhentheGree−n∞functionG(z ,r ;z ,r )
2 2 1 1
σq¯q(r1)Ψi(α,r1) maybeevaluatedanalyticallyiswhentheinitialandfinal
×
z sizes ofdipole are small, r r R (s). Inthis case
F (b, z) = dz dαd2r d2r d2r Ψ¯ (α,r ) | 1|∼| 2|≪ 0
2 ˆ 2 1 2 3 f 3 one can approximate in the rhs of Eq. (11)
G(+ ,~r ;z,~r )σ (r )G(z,~r ;z ,~r )
3 2 q¯q 2 2 2 1
× ∞
ρ (z ,b)σ (r )Ψ (α,r )
× A 2 q¯q 1 i 1 σq¯q(r) Cr2, (17)
≈
Equation (11) is quite complicated and in the general
case can be solved only numerically [56]. However in
some casesananalytic solutionis possible. For example, so the solution corresponds to the Green function of an
in the limit of a long coherence length, l R , rele- oscillator with a complex frequency,
c A
≫
vantforhigh-energyregion,onecanneglectthe “kinetic”
term ∆ G(z ,r ;z ,r ) in (11), and the Green func-
tion f∝ormar2lism 2rep2rod1uce1s the well-known eikonal for- a
G(z ,r ;z ,r )= (18)
2 2 1 1
mula in the “frozen” approximation [54]: 2πisin(ω∆z)
ia
G(z ,r ;z ,r ) = δ2(~r ~r ) (14) exp r2+r2 cos(ω∆z) 2~r ~r ,
2 2 1 1 2− 1 × (cid:18)2sin(ω∆z) 1 2 − 1· 2 (cid:19)
z2 (cid:2)(cid:0) (cid:1) (cid:3)
1
× exp−2σq¯q(r1)ˆ dζρA(ζ,b)
 z1 
so the amplitude (13) simplifies to
2iCρ
ω2 = − A ,
να(1 α)
1 −
a2 = iCρ να(1 α)/2.
(s,∆ ) = 2 d2bei∆~⊥·~b⊥ dαd2rΨ¯ (α,r)Ψ (α,r) − A −
A ⊥ ˆ ˆ f in ×
0
+∞ ThenforthefunctionsF wecanobtainexplicitexpres-
1 1,2
× 1−exp−2σq¯q(r) ˆ dζρA(ζ,b). (15)sions,
  −∞ 
5
1
a ia
F (b,z) = dαd2r d2r Ψ¯ (α,~r ) exp r2+r2 cos(ω∆z) 2~r ~r (19)
1 ˆ 1 2 f 2 2πisin(ω∆z) (cid:18)2sin(ω∆z) 1 2 − 1· 2 (cid:19)
(cid:2)(cid:0) (cid:1) (cid:3) ∆z=z∞−z
0
σ (~r ,s)Ψ (α,~r ),
q¯q 1 in 1
×
z 1
F (b,z) = dz dαd2r d2r d2r Ψ¯ (α,~r )σ (~r ,s)ρ (b,z )σ (~r ,s) (20)
2 ˆ 2ˆ 1 2 3 f 3 q¯q 2 A 2 q¯q 1
−∞ 0
a ia
exp r2+r2 cos(ω∆z) 2~r ~r
× 2πisin(ω∆z) (cid:18)2sin(ω∆z) 3 2 − 3· 2 (cid:19)
(cid:2)(cid:0) (cid:1) (cid:3) ∆z=z∞−z2
a ia
exp r2+r2 cos(ω∆z) 2~r ~r Ψ (α,~r ).
× 2πisin(ω∆z) (cid:18)2sin(ω∆z) 1 2 − 1· 2 (cid:19) in 1
(cid:2)(cid:0) (cid:1) (cid:3) ∆z=z2−z
InourevaluationsofG(z ,r ;z ,r )weusedanumer- In terms of the parton model one can interpret gluon
2 2 1 1
ical procedure discussed in detail in [56]. We would like shadowing as fusion of gluons originated from different
to emphasize that in contrastto our previous evaluation bound nucleons in the nucleus. Such a nonlinear effect
ofthepionproductiononaproton[6],inthispaperwedo leads to a reductionofthe gluondensity atsmall x com-
not introduce by hand the coherence lengths of the pion pared with an additive density [58–60]. While the quark
and effective axial meson state. They appear effectively distribution is directly measured in DIS, gluons can be
after convolution with proper distribution amplitudes. probed only via evolution, and this is why measurement
In addition to the coherent processes which leaves the ofgluonshadowingisstillachallenge. Theleadingorder
recoilnucleusintact,alargecontributiontopionproduc- analysis [61] based on the DGLAP evolution was found
tion comes from incoherent pion production, where the to be insensitive to gluon shadowing. Inclusion of data
target nucleus breaks up to fragments without particle on hadronic collisions made the analyses [62, 63] depen-
production, like in quasi-elastic scattering. In this case dent on debatable theoretical models, and led to such a
one can employ completeness of the final states, which stronggluonshadowingthattheunitaritybound[64]was
greatly simplifies the calculations. The analysis of such severely broken. The next-to-leading order fit [65] suc-
processesfor the electroproductionof vectormesons was ceeded to constrain the gluon shadowing correction at a
done in [57]. Its extension to the neutrino-production is rather small magnitude.
straightforwardand yields The theoretical predictions for gluon shadowing
strongly depend on the implemented model–while for
dσνA→lπA∗ = d2bdzeib·∆⊥ρ (b,z) F (b,z) F (b,z)2. x & 10−2 they all predict that the gluon shadowing is
dtdνdQ2 ˆ A | 1 − 2 | small or absent, for x . 10−2 the predictions vary in a
(21) wide range (see the review [66] and references therein).
Differently from the coherent case, the energy depen- Evaluationofthegluonshadowingwithinthecolordipole
denceofthecross-sectioniscontrolledonlybythecoher- model was performed in [55, 67]. In Fig. 1 the ratio of
ence length la, related to the heavy axial state. gluon distribution functions
c
g x,Q2
R ν,Q2 = A
B. Gluon shadowing g(cid:0) (cid:1) AgN(cid:0)(x,Q(cid:1)2)(cid:12)(cid:12)(cid:12)x=Q2/2mNν
(cid:12)
(cid:12)
As was mentioned above, the presence of higher Fock is plotted as function of energy ν for lead (A = 208).
componentscontaininggluonsleadstoanadditionalsup- We see that in the energy range ν . 100 GeV gluon
shadowing gives a small correction of the order of few
pression caused by multiple interactions of the gluons.
This suppression is called gluon shadowing. It is con- percent, so it can be safely neglected.
trolled by a new length scale lg, which turns out to
c
be much shorter than the coherence length for quarks.
As a result, no gluon shadowing is possible at Bjorken IV. RESULTS AND DISCUSSION
x > 10−2. Notice that at small Q2 Bjorken x is not a
proper variable, and one should switch to the energy de- In this section we present the results of numerical cal-
pendence. In this case the analog of gluon shadowing is culations. In this paper we make predictions in the
the Gribov inelastic shadowing correction [58] related to kinematics of the ongoing Minerva experiment at Fer-
triple-Pomeron diffraction. milab [1, 68]. While there are other experiments with
6
The plotted results also show that the plateau contracts
when Q2 increases, in agreement with coherence lengths
dependence on Q2 given in (9,10).
0.95 In the Figure 3 we compare the results for the ratio
1 d2σ /dνdQ2
) Rcoh (ν,Q2)= A , (23)
2Q A/N Ad2σN/dνdQ2
, 0.9
ν plotted by solid curves vs energy ν, with the expecta-
(
g
R tions based on the Adler relationshown by dashed lines.
OurresultsarebelowtheAdlerrelationpredictionsatall
0.85 Q2=1 GeV2 energies. As was explained in [6], at low energy the am-
plitudes of pion production on different nucleons are out
Q2=4 GeV2 ofcoherence,becausethe longitudinalmomentumtrans-
ferislarge. Athighenergies,accordingto[6],thelifetime
of the intermediate heavy states (a meson, ρπ, etc.) is
1 101 102 103 long, and absorptive corrections su1ppress the coherent
ν , GeV
cross section.
Thereis,however,awideenergyintervalfromfewhun-
Figure 1: Gluon shadowing as a function of energy. See [55,
dredsMeVupto about10GeV, wherethe Adlerrelation
67] for details of evaluation.
was predicted to be at work [6]. Now we see that even
at these energies the Adler relation is broken. To un-
derstand why this happens notice that an effective two-
energiesevenhigherthan forMinerva,they havea much
channel model used in [6] assumed dominance of two
worse statistics [69, 70].
statesinthe dispersionrelationforthe axialcurrent,the
In Fig. 2 the ratio of the cross-sections on nuclei and
pionandaneffectiveaxialvectorpoleawiththemassof
nucleon,
theorderof1GeV. TheconditionofvalidityoftheAdler
dσ /dtdνdQ2 relation was shortness of the coherence length related to
Rcoh (t,ν,Q2)= νA→lπA , (22)
A/N A2dσ /dtdνdQ2 the mass of the a-state compared to the nuclear size,
νN→lπN
2ν
is plotted as function of transferred energy ν. In the la = R . (24)
same Figure we plotted with dashed lines predictions of c Q2+m2 ≪ A
a
the Adler relation. As was discussed in Sect. III, diffrac-
In contrast to the simple model, the invariant mass of
tive pion production on nuclei is characterized by three
a q¯q dipole is not fixed, m2 = (m2 + k2)/α(1 α),
physicallydistinct energyintervals,controlledby the co- q¯q q T −
where α is the fractional light-cone momentum of the
herence lengths related to the masses of pion and heavy
quark. Correspondingly, the related coherence length,
axial states. Indeed, one can see in the left pane, the
lq¯q is distributed over a wide mass range, and while the
cross-sectionhas threedifferentregimes. The lowenergy c
centerofthedistributionandlargemassesleadtoashort
region, ν . 1 GeV, is controlled by the pion coherence
lq¯q,thelow-masstailofthisdistributionresultsinalong
length, and the cross section is suppressed if lπ is short. c
c lq¯q R . For this reason the absorption corrections
Notice,however,thatattheselowenergiesthedipolede- c ≫ A
suppress the cross section, as we see in Fig. 3, even at
scription is rather formal, because one should take into
moderate energies.
account the resonances like was done in [15, 16].
InFig.4theratiooftheincoherentnuclear-to-nucleon
Intheregion1.ν .10GeVthefinalq¯qdipole(pion)
cross-sections
is produced momentarily, and the absorptive corrections
suppressthecrosssectionofneutrino-productionofpions Rinc (t,ν,Q2)= dσνA→lπA∗/dtdνdQ2 , (25)
qualitatively in the same way as shadowing does in the A/N Adσ /dtdνdQ2
νN→lπN
pion-nucleus elastic cross section. However, as will be
discussedbelow,duetothefactthatincolordipolemodel is plotted versus energy ν. As was discussed in Sec-
thereisnoexplicitaxialmesonstates,thecross-sectionis tion III, the energy dependence of the incoherent cross-
uptothirtypercentlessthanthe predictiongivenbythe sectioniscontrolledonlybytheshortestcoherencelength
Adler relation (1). This issue will be discussed in detail la,relatedtotheheavyaxialstates,sothereareonlytwo
c
a few paragraphs below. regimes: la R and la > R . Our numerical calcula-
c ≤ A c A
In the regionν &10GeV allthe coherence time scales tions confirm such a behavior.
becomelong,sotheq¯qdipoleisproducedlonginadvance Interesting that in this case of incoherent pion pro-
of the interaction and propagates through the whole nu- duction the Adler relation turns out to be severely bro-
cleus. In this case the absorptive corrections reach the ken at low energy, but is restored at high energies when
maximal strength and suppress the cross section consid- la R , i.e. demonstrating a trend opposite to coher-
c ≫ A
erably. One can see that in the right pane of Fig. 2. ent production. Indeed, according to the Adler relation
7
Q2=0 GeV2, ∆ =0
⊥
Q2=0.2 GeV2, ∆ =0
0.6 ⊥
Q2=0.5 GeV2, ∆ =0
0.4 ⊥
Q2=1 GeV2, ∆ =0
⊥
0.5 Q2=2 GeV2, ∆ =0
⊥
0.3
0.4
hN hN
coA/ coA/
R R
0.3
0.2
0.2
A=12, ∆⊥=0 0.1
0.1 A=55, ∆ =0
⊥
A=208, ∆ =0
⊥
1 10 1 10
ν , GeV ν , GeV
Figure 2: [Color online] ν-dependence of the ratio of the coherent forward neutrino-pion production cross-sections on nuclear
and proton targets. Left: ν-dependence of the ratio for different nuclei at Q2 = 0. Solid curves correspond to the color
dipole model, dashed lines show thepredictions of the Adler relation. Right: ν-dependenceof thenuclear ratio vs Q2 for lead
(A=208).
0.9 Q2=0 Q2=1 GeV2 Q2=2 GeV2
0.8 A=12
Pb
0.7
1 Al 0.6 A=55
cN
cohRA/N C inRA/ 00..54
A=208
0.5 0.3
0.2
0.1
0 1 101 102
-1 2
10 1 ν (GeV)10 10 ν , GeV
Figure3: [Coloronline]Comparisonofthenucleareffectspre-
dictedin [6](dashedcurves)andin thispaper(solid curves). Figure 4: [Color online] ν-dependence of the ratio of the in-
coherent forward pion neutrino-production cross-sections on
nuclear and proton targetsat different virtualities Q2.
thecrosssectionofincoherentpionproductionispropor-
tional to the cross section of quasi-elastic pion-nucleus
ent cross-section surpasses the coherent one.
scattering πA πA∗, where the projectile pion must
→
propagate and survive through the whole nuclear thick-
ness. The same occurs with the q¯q dipole in the case of
la R . V. SUMMARY
c ≫ A
Comparison of Figs. 2 and 4 shows that in the for-
ward kinematics (∆ = 0) the coherent cross-section is We performed calculations for the nuclear effects in
⊥
much higher than the incoherent one. However, for the diffractive neutrino-production of pions basing on the
off-forward case the coherent cross-section is suppressed color-dipole description. The non-zero phase shifts be-
by the nuclear formfactor, whereas the incoherent cross- tween the production amplitudes on different bound nu-
section is controlled by the proton formfactor. For this cleons are taken into account applying the path integral
reason,forsufficientlylargevaluesoft=∆2 theincoher- technique. The results confirmed the presence of promi-
8
nentstructuresintheenergydependenceofcoherentand sumed in the 2-channelmodel. The contribution of such
incoherentprocessesonnuclei. Althoughthegeneralpat- light dipoles is subject to strong absorptive corrections
tern of nuclear effects agrees with what was predicted in reducing the cross section. At the same time, at high
[6] within a simple 2-channel model, the new important energiesbothmodels predicta similarstrongbreakdown
featuresareobservedbasingonthemoredetaileddynam- of the Adler relation.
ics of the dipole description. Namely, while the effective
2-channel model predicted validity of the Adler relation
in the wide energy interval from few hundreds MeV up
to about 10GeV [6], with the dipole approach we found
Acknowledgments
a considerable suppression of the nuclear cross section
compared to the result of the Adler relation within the
same energy interval. This happens, due to a contribu- This work was supported in part by Fondecyt (Chile)
tion of light dipoles possessinga longer coherencelength grants 1090291, 1100287 and 1120920, and by Conicyt-
comparedto the fixed mass heavy intermediate state as- DFG grant No. 084-2009.
[1] D. Drakoulakos et al. [Minerva Collaboration], ph]].
arXiv:hep-ex/0405002. [24] B. Z. Kopeliovich, I. Schmidt and M. Siddikov,
[2] S.L. Adler, Phys.Rev. 135 (1964) B963. Phys. Rev. D 82 (2010) 014017 [arXiv:1005.4621 [hep-
[3] S.L.AdlerandY.Dothan,Phys.Rev.151(1966) 1267. ph]].
[4] A. A. Belkov and B. Z. Kopeliovich, Sov. J. Nucl. Phys. [25] B. Z. Kopeliovich, I. Schmidt and M. Siddikov,
46, 499 (1987) [Yad.Fiz. 46, 874 (1987)]. Phys. Rev. D 81 (2010) 094013 [arXiv:1003.4188 [hep-
[5] D.Rein andL.M.Sehgal, Nucl.Phys.B223(1983) 29. ph]].
[6] B. Z. Kopeliovich, I. Potashnikova, M. Siddikov, [26] M. V. T. Machado, Eur. Phys. J. C 59 (2009) 769
I. Schmidt, Phys. Rev. C 84 (2011) 024608 [arXiv:0810.3665 [hep-ph]].
[arXiv:1105.1711 [hep-ph]]. [27] M. V. T. Machado, Phys. Rev. D 78 (2008) 034016
[7] J. Bell, Phys. Rev.Lett.13 57 (1964). [arXiv:0805.3144 [hep-ph]].
[8] R. J. Glauber, in Lectures in Theoretical Physics, W. E. [28] M. V.T. Machado, arXiv:0905.4516 [hep-ph].
Brittin et al Editors, NewYork (1959). [29] R. Fiore and V. R. Zoller, Phys. Lett. B 632 (2006) 87
[9] V. N. Gribov, Sov. Phys. JETP 29 (1969) 483 [arXiv:hep-ph/0509097].
[Zh.Eksp. Teor. Fiz. 56 (1969) 892]. [30] R. Fiore and V. R. Zoller, JETP Lett. 82 (2005) 385
[10] B. Z. Kopeliovich, Phys.Lett. B 227, 461 (1989). [Pisma Zh. Eksp. Teor. Fiz. 82 (2005) 440] [arXiv:hep-
[11] B. Z. Kopeliovich, Nucl. Phys. Proc. Suppl. 139, 219 ph/0508187].
(2005). [31] R. Fiore and V. R. Zoller, JETP Lett. 87 (2008) 524
[12] B. Z. Kopeliovich, Sov. Phys. JETP 70 (1990) 801 [Zh. [arXiv:0803.4492 [hep-ph]].
Eksp.Teor. Fiz. 97 (1990) 1418]. [32] R. Fiore and V. R. Zoller, Phys. Lett. B 681 (2009) 32
[13] WA59 Collaboration, P.P. Allport et al., Phys. Lett. B [arXiv:0812.4501 [hep-ph]].
232 417 (1989). [33] M. B. Gay Ducati, M. M. Machado and
[14] B. Z. Kopeliovich, L. I. Lapidus and A. B. Zamolod- M. V. T. Machado, Phys. Rev. D 79 (2009) 073008
chikov,JETPLett.33,595(1981)[PismaZh.Eksp.Teor. [arXiv:0812.4273 [hep-ph]].
Fiz. 33, 612 (1981)]. [34] M. B. Gay Ducati, M. M. Machado and
[15] O. Lalakulich, E. A. Paschos and G. Piranishvili, M. V.T. Machado, Braz. J. Phys.38 (2008) 487.
Phys.Rev.D 74 (2006) 014009 [arXiv:hep-ph/0602210]. [35] M.B.G.Ducati,M.M.MachadoandM.V.T.Machado,
[16] O. Lalakulich, W. Melnitchouk and E. A. Paschos, Phys. Lett. B 644 (2007) 340 [arXiv:hep-ph/0609088].
Phys.Rev.C 75 (2007) 015202 [arXiv:hep-ph/0608058]. [36] M. V. T. Machado, Phys. Rev. D 75 (2007) 093008
[17] S.X. Nakamura,arXiv:1109.4443 [nucl-th]. [arXiv:hep-ph/0703111].
[18] K. J. Golec-Biernat and M. Wüsthoff, Phys. Rev. D 59 [37] B. Z. Kopeliovich, I. Schmidt and M. Siddikov,
(1999) 014017 [arXiv:hep-ph/9807513]. Phys. Rev. D 84 (2011) 033012 [arXiv:1107.2845 [hep-
[19] K. J. Golec-Biernat, Acta Phys. Polon. B 35, 3103 ph]].
(2004). [38] A. E. Dorokhov, W. Broniowski and E. Ruiz Arriola,
[20] J.Raufeisen,J.-C.Peng,G.C.Nayak,Phys.Rev.D66, Phys.Rev.D 74 (2006) 054023 [arXiv:hep-ph/0607171].
034024 (2002). [hep-ph/0204095]. [39] I. V. Anikin, A. E. Dorokhov and
[21] J. Hufner, Yu. P. Ivanov, B. Z. Kopeliovich and L. Tomio, Phys. Part. Nucl. 31 (2000) 509
A.V.Tarasov,Phys.Rev.D62(2000)094022[arXiv:hep- [Fiz. Elem. Chast. Atom. Yadra31 (2000) 1023].
ph/0007111]. [40] A. E. Dorokhov and W. Broniowski, Eur. Phys.J. C 32
[22] B. Z. Kopeliovich, I. Schmidt and M. Siddikov, (2003) 79 [arXiv:hep-ph/0305037].
Phys. Rev. D 79 (2009) 034019 [arXiv:0812.3992 [hep- [41] K. Goeke, M. M. Musakhanov and M. Siddikov,
ph]]. Phys. Rev. D 76 (2007) 076007 [arXiv:0707.1997 [hep-
[23] B. Z. Kopeliovich, I. Schmidt and M. Siddikov, ph]]
Phys. Rev. D 80 (2009) 054005 [arXiv:0906.5589 [hep- [42] B. Z. Kopeliovich, I. Schmidt and M. Siddikov,
9
arXiv:1108.5654 [hep-ph]. [57] B. Z. Kopeliovich, J. Nemchik, A. Schäfer and
[43] N. H. Buttimore, B. Z. Kopeliovich, E. Leader, J. Sof- A. V. Tarasov, Phys. Rev. C 65 (2002) 035201
fer and T. L. Trueman, Phys. Rev. D 59 (1999) 114010 [arXiv:hep-ph/0107227].
[arXiv:hep-ph/9901339]. [58] L.V.Gribov,E.M.LevinandM.G.Ryskin,Phys.Rept.
[44] B. Z. Kopeliovich, H. J. Pirner, A. H. Rezaeian 100 (1983) 1.
and I. Schmidt, Phys. Rev. D 77 (2008) 034011 [59] O. V. Kancheli, Pisma Zh. Eksp. Teor. Fiz. 18 (1973)
[arXiv:0711.3010 [hep-ph]]. 465.
[45] B. Z. Kopeliovich, A. H. Rezaeian and I. Schmidt, [60] A. H. Mueller and J. -w. Qiu, Nucl. Phys. B 268, 427
Phys. Rev. D 78 (2008) 114009 [arXiv:0809.4327 [hep- (1986).
ph]]. [61] K. J. Eskola, V. J. Kolhinen and C. A. Salgado,
[46] B. Z. Kopeliovich, I. K. Potashnikova, I. Schmidt Eur. Phys. J. C 9 (1999) 61 [arXiv:hep-ph/9807297].
and J. Soffer, Phys. Rev. D 78 (2008) 014031 [62] K. J. Eskola, H. Paukkunen and C. A. Salgado, JHEP
[arXiv:0805.4534 [hep-ph]]. 0807, 102 (2008) [arXiv:0802.0139 [hep-ph]].
[47] J. B. Bronzan, G. L. Kane and U. P. Sukhatme, [63] K. J. Eskola, H. Paukkunen and C. A. Salgado, JHEP
Phys.Lett. B 49 (1974) 272. 0904 (2009) 065 [arXiv:0902.4154 [hep-ph]].
[48] J. D. Bjorken and J. B. Kogut, Phys. Rev. D 8, 1341 [64] B. Z. Kopeliovich, E. Levin, I. K. Potashnikova and
(1973). I.Schmidt,79,064906(2009)[arXiv:0811.2210[hep-ph]].
[49] B. Kopeliovich and B. Povh, Phys. Lett. B 367, [65] D. de Florian, R. Sassot, M. Stratmann and
329 (1996); Z. Phys. A 356, 467 (1997) [arXiv:nucl- W. Vogelsang, Phys. Rev. Lett. 101 (2008) 072001
th/9607035]. [arXiv:0804.0422 [hep-ph]].
[50] B. Z. Kopeliovich, I. K. Potashnikova, I. Schmidt and [66] N. Armesto, J. Phys. G 32 (2006) R367 [arXiv:hep-
J. Soffer, arXiv:1109.2500 [hep-ph]; to appear in Phys. ph/0604108].
Rev.D. [67] B. Z. Kopeliovich, J. Raufeisen, A. V. Tarasov and
[51] R.T. Deck, Phys.Rev.Lett. 13, 169 (1964). M. B. Johnson, Phys. Rev. C 67 (2003) 014903
[52] B. Z. Kopeliovich and P. Marage, Int. J. Mod. Phys. A [arXiv:hep-ph/0110221].
8, 1513 (1993). [68] K. S. McFarland [MINERvA Collaboration],
[53] B. Z. Kopeliovich, J. Raufeisen and A. V. Tarasov, 440 Nucl. Phys. Proc. Suppl. 159 (2006) 107
(1998) 151 [arXiv:hep-ph/9807211]. [arXiv:physics/0605088].
[54] B.Z.KopeliovichandB.G.Zakharov,Phys.Rev.D44, [69] J. Bell et al.,Phys.Rev. Lett.41 (1978) 1008.
3466 (1991). [70] P. Allen et al. [Aachen-Birmingham-Bonn-CERN-
[55] B. Z. Kopeliovich, A. Schäfer and A. V. Tarasov, Phys. London-Munich-Oxford Collaboration], Nucl. Phys. B
Rev.D 62, 054022 (2000). 264 (1986) 221.
[56] J.Nemchik,Phys.Rev.C68,035206 (2003) [arXiv:hep-
ph/0301043].