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Transverse polarization distribution and fragmentation functions

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1Transverse polarization distribution and fragmentation functions ?)Dani¨e l Boer RIKEN-BNL Research Center Brookhaven National Laboratory,Upton,NY 11973,U.S.A.(Received February 1,2008)We discuss transverse polarization distribution and fragmentation functions,in particu-lar,T-odd functions with transverse momentum dependence,which might be relevant for the description of single transverse spin asymmetries.The role of intrinsic tranverse momentum in the expansion in inverse powers of the hard scale is elaborated upon.The sin φsingle spin asymmetry in the process e p →e ′π+X as recently reported by the HERMES Collaboration is investigated,in particular,by using the bag model.§1.Introduction Transverse polarization distribution and fragmentation functions parameterize transverse spin e?ects in hard scattering processes.Although in general little is known experimentally on most of these functions,some transverse spin experiments have been performed.For instance,large single transverse spin asymmetries have been observed 1)in the process p p ↑→πX .Therefore,the question arises which transverse polarization distribution and/or fragmentation functions are relevant for their description.This might include so-called T-odd fragmentation functions,which are expected to arise due to ?nal state interactions.The main observation is that the description of this speci?c process in terms of transverse spin functions will lead to power suppressed single spin asymmetries,unless one takes into account the intrinsic transverse momentum of the partons,see e.g.Ref.2).We will discuss how the transverse momentum dependence of asymmetries can contain information (in terms of distribution and fragmentation functions)at leading order,that would be power suppressed if integrated over the transverse momentum.

Especially the T-odd functions with transverse momentum dependence might be relevant for the description of single transverse spin asymmetries,since these functions link the transverse momentum and transverse spin (of either quarks or hadrons)with a speci?c handedness.The di?erent functions will lead to di?erent angular dependences.Hence,studying the angular dependences of asymmetries (and their transverse momentum dependence)is a most promising way to unravel the origin(s)of transverse spin asymmetries.This is for instance demonstrated by a recent result by the HERMES Collaboration 3),as will be discussed.

2Dani¨e l Boer

§2.Distribution and fragmentation functions Transverse spin asymmetries in hadron-hadron collisions require an explanation that involves quarks and gluons.A large scale allows for a factorization of such processes into parts describing the soft physics convoluted with a hard subprocess cross section.We will?rst focus on the Drell-Yan(DY)process,i.e.lepton pair production.In lowest order–the parton model approximation–this process consists of two soft parts,the correlation functions calledΦand

2

[f1(x)P+g1(x)λγ5P+h1(x)γ5S T P].(2.1) Other common notation is q for f1,?q for g1andδq or?T q for h1.

At the parton model level one?nds the well-known double transverse spin asym-metry4),A T T∝|S1T||S2T|cos(φS1+φS2)h1(x1)

?s which are expected to be small.The higher twist corrections require parameterizing the correlation functionΦ(x)to include contribu-tions proportional to the hadronic scale(say the hadron mass),

Φ(x)=Eq.(2.1)+M

2

γ5[n+,n?] .(2.2)

The distribution functions e,g T,h L are so-called twist-3functions;these will show up in the cross section suppressed by M/Q,where Q is a hard scale.At leading order in

Transverse polarization distribution and fragmentation functions3αs,i.e.(αs)0,but at the order1/Q,one?nds6),7)no single or double transverse spin asymmetries in the DY cross section.The functions g T and h L will only?)appear6) in the asymmetry A LT.

Therefore,in order to produce a large single transverse spin asymmetry in the DY process(no experimental data exists however),one needs some conceptually non-trivial mechanism,like soft gluon poles,since regular perturbative and higher twist contributions appear to be either small or absent.For the case of pion production in p p↑scattering there exist a more conventional explanation of the large observed single spin asymmetries.It involves a T-odd fragmentation function.Such functions are expected to be present even if time reversal invariance is applied,because of the?nal state interactions between the outgoing hadron and the other fragments. For the description of a quark fragmenting into a pion plus anything,one needs the fragmentation correlation function8)?(z),which one parameterizes in accordance with the required symmetries(including time reversal)

?(z)=T-even part+M

2

[n?,n+] .

The twist-3fragmentation functions D T,E L,H are T-odd.The T-odd fragmentation function H can be responsible for the single spin asymmetries,e.g.via a diagram as depicted in Fig.2.This yields power suppressed asymmetries,which can be investigated experimentally by changing the energy scale over a wide range(again RHIC can provide this information).The question we like to address here is:how to obtain a single spin asymmetry that is not suppressed by powers of the hard scale?A possible solution is to include intrinsic transverse momentum4),i.e.replace Φ(x)→Φ(x,p T)and?(z)→?(z,k T).For the T-even distribution functions in DY this replacement leads to double spin asymmetries4),7).But T-odd functions with transverse momentum dependence can lead to single spin asymmetries at leading order.The transverse momentum dependent distribution functions are de?ned as

Φ(x,p T)=1

M ,(2.3)

where f1=f1(x,p T),etc.and we use the shorthand notation (..)1s(x,p T)≡λ(..)1L(x,p2T)+

(p T·S T)

2 D⊥1T?μνρσγμPνkρT SσT M .(2.5)

The fragmentation functions D⊥1T and H⊥1are T-odd functions and as can be seen in Figs.3and4such T-odd e?ects link transverse momentum and transverse spin (orthogonal to the transverse momentum)with a speci?c orientation(handedness). The chiral-even function9)D⊥1T is expected to be relevant for transversely polarized

4Dani¨e l Boer

-

T

⊥D 1T =Fig.3.The chiral-even,T-odd function D ⊥1T signals di?erent probabilities for q →Λ(k T ,±S T )+X .

Λproduction 10),for instance in p p →Λ↑X (also measurable at RHIC).The chiral-odd function 11)H ⊥1,also called the “Collins e?ect”function,might be relevant for p p ↑→πX asymmetry via 12):A T ~h 1(x 1)?f 1(x 2)?H ⊥1(z,k T

).Like the transverse

-H ⊥1=

Fig.4.The Collins e?ect function H ⊥1signals di?erent probabilities for q (±S T )→π(k T )+X .

momentum of the lepton pair in the DY process,the transverse momentum of the pion now originates from the intrinsic transverse momentum of the initial partons in addition to transverse momentum generated perturbatively by radiating o?some additional parton(s)in the ?nal state (hence the transverse momentum of the pion need not be small).

There is an experimental indication 13)from analyzing a particular angular de-pendence (a cos 2φdependence 14))in the unpolarized process e +e ?→Z 0→ππX ,where the pions belong to opposite jets,that the Collins e?ect is in fact a few percent of the magnitude of the ordinary unpolarized fragmentation https://www.sodocs.net/doc/7418184965.html,parison to the magnitude at lower energies of course requires evolution.

The two T-odd fragmentation functions satisfy the following sum rules 15)?):

h dz z H ⊥(1)1(z )=0,

h dz z D ⊥(1)1T (z )=0,(2.6)

where F (1)(z )=z 2 d 2k T k 2T /(2M 2)F (z,z 2k 2T ).The function D ⊥1T can also be probed in charged current exchange processes,since it is chiral-even,as opposed to the chiral-odd functions like h 1,H ⊥1.So far we have not commented on the fact that the T-odd fragmentation fuctions

appearing in the parameterization of ?(z )will lead to power suppressed contribu-tions,whereas the two T-odd functions of ?(z,k T )can contribute at leading order in the expansion in inverse powers of the hard scale.This is the subject of the next section.§3.Transverse momentum at leading “twist”

A process like deep inelastic scattering has only two scales,the hadronic scale (say the hadron mass M )and the large scale of the virtual photon (Q ).The explicit

Transverse polarization distribution and fragmentation functions5 mass term in front of the function g T inΦ(x)Eq.(2.2)can therefore only lead to a term M/Q in the cross section.

In the case of a less inclusive experiment,for instance e p→e′πX,one can also observe the transverse momentum Pπ⊥of the pion.This semi-inclusive cross section will depend on three dimensionful quantities:M,|Pπ⊥|and Q.The function g T will again lead to a contribution~M/Q,since it is not sensitive to the transverse

momentum of the pion;it will not average to zero if one averages over Pπ⊥.On the other hand,one might be sensitive to functions that will disappear upon averaging. For these functions the appropriate scales are M and Pπ⊥.Terms proportional to M/|Pπ⊥|will appear without expanding in M/|Pπ⊥|.We will see an explicit example below.

In order to arrive at a single transverse spin asymmetry that is not suppressed by inverse powers of the hard scale,one can consider cross sections di?erential in the transverse momentum of the pion.In that case one is sensitive to the transverse momentum of the quarks directly and in case this concerns intrinsic transverse mo-mentum of the quarks inside a hadron,the e?ects need not be suppressed by1/Q. The point is that if the transverse momentum of the pion is(solely)produced by perturbative QCD corrections,each factor of transverse momentum has to be ac-companied by the inverse of the scale in the elementary hard scattering subprocess: 1/Q.But in case of intrinsic transverse momentum of the quarks(or gluons)the relevant scale is not Q,but the hadronic scale M.In other words,one is not allowed to make an expansion ofΦ(x,p T)in terms of p T/Q,since there could be terms of order p T/M,which are not small in general.

In processes with two(or more)soft parts,like semi-inclusive leptoproduction, the intrinsic transverse momentum of one soft part is linked to that of the other soft part resulting in e?ects,e.g.azimuthal asymmetries,not suppressed by1/Q.These e?ects will show up at relatively low(including nonperturbative)values of|Pπ⊥|.By radiating o?hard partons the fragmenting quark and hence the pion might achieve a higher transverse momentum.Of course,if this produces a very high transverse momentum(~Q),then this will lead to suppression.But the point is:in case one observes the transverse momentum of the pion,one can probeΦ(x,p T)and?(z,k T), without suppression by1/Q.

Let us investigate the example16)of e p→eπX and consider cross sections integrated,but weighted with a function of the transverse momentum of the pion:

P e P p

W P e P p= dφe d2Pπ⊥W dσ[ e p→eπX]

6Dani¨e l Boer

now restrict to jet production(i.e.,keeping only D1(z)=δ(1?z))and consider pion production in the next section.

For the case of P e P p=LT we?nd the following power suppressed azimuthal spin asymmetry

1 LT1?y M

[4πα2s/Q4]

=cosφe Sλe|S T|y(1?1

2 g(1)1T(x)SαTγ5P?h⊥(1)1L(x)λγ5γαT P .(3.4)

Clearly

the factor M has to be compensated in the cross section and as is seen

from Eq.(3.3)the relevant scale is1/ |P jet⊥| .Nevertheless,from the QCD e.o.m. it is clear thatΦα?(x)is related to a quark-gluon-quark matrix element that will always show up M/Q suppressed in DIS.And in fact,the function g(1)1T is a well-known quantity in the Wandzura-Wilczek approximation:it equals(upon neglecting

quark masses)x g W W

T (x),where g W W

T

=g1+g W W

2

.The existing data on g T are

(still)consistent with g T=g W W

T .This shows that part of the“twist-3”information

(the dominant contribution in fact)can be obtained without suppression factors of1/Q,by doing a less inclusive experiment,namely by observing a transverse momentum.Of course the average transverse momentum is also a function of Q and experimentally the observation of transverse momenta is more di?cult,but the

point remains that the same information(e.g.g W W

T )can enter with di?erent scales

in di?erent quantities.

§4.Single spin asymmetry in e p→e′π+X

Recently,the HERMES Collaboration3)reported a sinφasymmetry in the pro-cess e p→e′π+X,where the target has a polarization along the electron beam direction andφ(=φeπ)is the angle of the transverse momentum of the pion with respect to the lepton scattering plane,cf.Fig.5.The measured asymmetry has an analyzing power of0.022±0.005±0.003.The fact that the pion prefers to be out-of-plane in the asymmetry(a sinφdistribution)is indicative of a T-odd e?ect.The asymmetry can indeed be expressed in terms of a chiral-odd,T-odd fragmentation function with transverse momentum dependence,the Collins e?ect function H⊥1,if a factorized picture like in Ref.9)is assumed:

Wμν= d2p T d2k Tδ2(p T+q T?k T)Tr Φ(x,p T)γμ?(z,k T)γν .(4.1)

Transverse polarization distribution and fragmentation functions7

Fig.5.Kinematics of the sinφasymmetry in semi-inclusive deep inelastic scattering. One can easily project out the sinφdependence from the cross section by weighted

integration.We de?ne A P

e P p = W P

e P p

/ 1 OO with W=(|Pπ⊥|/zMπ)sinφeπ.

We will assume the asymmetry arises mainly from the dominant?avor,i.e.we take into account only the contribution from u→π+.Furthermore,we will neglect transverse momentum e?ects inside the proton.We?nd the following expressions for the relevant asymmetries:

A OL∝λ2(2?y) Q x h L(x)H⊥(1)1(z),(4.2)

A OT∝|S T|(1?y)h1(x)H⊥(1)1(z),(4.3)

where H⊥(1)

1(z)=z2 d2k T k2T/(2M2π)H⊥1(z,z2k2T).

The polarization of the target P p is in the lepton scattering plane and in fact along the electron beam direction,hence,it is a combination of L and T,depending on y=(P·q)/(P·l).We?nd

|S T|1?y2Mx

A OT =

2?y

h1(x)

.(4.4)

A striking feature is that the contribution from the target spin transverse to the virtual photon momentum also enters at subleading order in M/Q,even though the relevant functions(h1and H⊥1)are leading twist functions.For0.2

then?nds

2h L

A OT

<~6h L

8

Dani¨e l Boer 00.5

11.5

2

00.20.40.60.81x

h 1h L

h L WW

Fig.6.Bag model functions h 1,h L and h W W L .00.511.522.500.10.20.30.40.50.6

0.70.8x h L /h 1

Fig.7.Bag model ratio h L /h 1.

contribute M 2/Q 2suppressed as one can see by interchanging the role of |S T |and λin Eq.(4.4).

The asymmetry A LO has also been measured and was found to be consistent with zero,coinciding with the Wandzura-Wilczek expectation:

A LO ∝λe 2y Q x e (x )H ⊥(1)1(z )W W ∝m u

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