Gauge invariance and gluon poles for direct photon production

0.22 MB·English

by  I. V. Anikin

Gauge invariance and gluon poles for direct photon
production
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n I.V. Anikin∗
a BogoliubovLab. ofTheoreticalPhysics,JINR,141980Dubna,Russia
J
E-mail: [email protected]

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2 O.V. Teryaev

BogoliubovLab. ofTheoreticalPhysics,JINR,141980Dubna,Russia
]
h E-mail: [email protected]
p
-
p
e Wediscussthehadrontensorofthedirectphotonproduction. Westudytheeffectswhichleadto
h
thesoftbreakingoffactorizationbyinspectionofthecorrespondingQCDgaugeinvariance. We
[

emphasizethatthespecialroleisplayedbythecontourgaugeforgluonfields. Wedemonstrate
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v that the different prescriptions in the gluonic pole contributions are dictated by the presence of
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initialorfinalstateinteractionsindiagrams.Moreover,thedifferentprescriptionsthatcorrespond
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9 totheinitialoffinalstateinteractionsareneededtoensuretheQCDgaugeinvariance.
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i
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r
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XXIIInternationalBaldinSeminaronHighEnergyPhysicsProblems
15-20September,2014
JINR,Dubna,Russia
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Gaugeinvarianceandgluonpolesfordirectphotonproduction I.V.Anikin
1. Introduction
As shown in [1], to ensure the QED gauge invariance of the transverse polarized Drell-Yan
(DY) hadron tensor it is a must to include a contribution of the extra diagram which arises from
the non-trivial imaginary part of the corresponding twist 3 function BV(x ,x ). Previously, how-
1 2
ever, this function assumed to be a real function (see, for example, [2] where, nevertheless, the
needed imaginary part was generated by the specially introduced “propagator” in the hard part
of the hadron tensor). As explained in [1], the complex prescription in the representation of BV-
functioncanbeunderstoodwithahelpofthecontourgauge. Moreover, theaccountforthisextra
contribution, owing to the complex BV-function, led to the amplification of the hadron tensor by
thefactorof2. Noticethatthisourfindingwasindependentlyconfirmedin[3]byusingofthedif-
ferent approach. The corresponding SSAs, related to the use of the twist 3 BV-function in the DY
process,andtheroleofgluonpoleswerepreviouslydiscussedbymanygroups(see,forexample,
[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]).
Wenowpresentourapproach,thatwasusedin[1]andrecentlydevelopedin[19],tostudythe
effects in the direct photon production (DPP) which lead to the soft breaking of factorization (or
theuniversalitybreaking)byinspectionoftheQCDgaugeinvariance. Asin[1],thespecialroleis
played by the contour gauge for gluon fields. We, first, demonstrate that the prescriptions for the
gluonic poles in the twist 3 correlators are dictated by the prescriptions in the corresponding hard
partsand,second,arguethatthedifferentprescriptionsinthegluonicpolesdefinedbytheinitialor
finalstateinteractionsinthediagramsunderconsideration(see, [19]formoredetails). Moreover,
the different prescriptions in the representations of BV-functions are needed to ensure the QCD
gaugeinvariance. Thesituationwhenwehavenotheuniversalityconditionforthecorresponding
softfunctionwillbetreatedasoftbreakingoffactorization.
2. Kinematics
Westudythesemi-inclusiveprocesswherethehadronwiththetransversepolarizationcollides
withtheotherunpolarizedhadrontoproducethedirectphotoninthefinalstatein:
N(↑↓)(p )+N(p )→γ(q)+q(k)+X(P ). (2.1)
1 2 X
For (2.1) (also for the Drell-Yan process), the gluonic poles manifest [12]. We perform our cal-
culations within a collinear factorization and, therefore, it is convenient (see, e.g., [21]) to fix the
dominantlight-conedirectionsas
(cid:114) (cid:114)
S S
p = n∗, p = n,
1 2
2 2
√ √ √ √
n∗ =(1/ 2,0 ,1/ 2), n =(1/ 2,0 ,−1/ 2). (2.2)
µ T µ T
Accordingly,thequarkandgluonmomentak and(cid:96)liealongtheplusdominantdirectionwhilethe
1
gluonmomentumk –alongtheminusdirection. Thefinalon-shellphotonandquark(anti-quark)
2
momentacanbepresentedas
(cid:114) (cid:114)
S q2 S k2
q=y n− √⊥ n∗+q , k=x n∗− √⊥ n+k . (2.3)
B ⊥ B ⊥
2 y 2S 2 x 2S
B B
2
Gaugeinvarianceandgluonpolesfordirectphotonproduction I.V.Anikin
p1 p1
p2 p2
Figure1: TheFeynmandiagramdescribingthehadrontensorofthedirectphotonproduction.
And,theMandelstamvariablesfortheprocessandsubprocessaredefinedas
S=(p +p )2, T =(p −q)2, U =(q−p )2,
1 2 1 2
sˆ=(x p +yp )2=x yS, tˆ=(x p −q)2=x T, uˆ=(q−yp )2=yU. (2.4)
1 1 2 1 1 1 1 2
Theamplitude(orthehadrontensor)of(2.1)constructedbythecontributionsfrom(i)theleading
(LO) diagrams: two diagrams with a radiation of the photon before (ALO) and after (ALO) the
1 2
quark-gluonvertexwiththegluongoingtothelowerblob,seetherightsideofFig.1;(ii)thenext-
to-leadingorder(NLO)diagrams: eightdiagramsconstructedfromtheLOdiagramsbyinsertionof
allpossibleradiationsoftheadditionalgluonwhichtogetherwiththequarkgoestotheupperblob,
seetheleftsideofFig.1. So,wehavethehadrontensorrelatedtothecorrespondingasymmetry:
2 8
dσ↑−dσ↓∼W = ∑ ∑ALO∗BNLO. (2.5)
i j
i=1j=1
Here, we will mainly discuss the hadron tensor rather than the asymmetry itself. So, the hadron
tensor as an interference between the LO and NLO diagrams, ALO∗BNLO, can be presented by
i j
Fig. 1 where the upper blob determines the matrix element of the twist-3 quark-gluon operator
while the lower blob – the matrix element of the twist-2 gluon operator related to the unpolarized
gluondistribution.
3. Factorizationprocedure
Thecollinearfactorizationbeingourmaintool,letusoutlinethemainstepsofthefactorization
procedure. Itcontains(i)thedecompositionofloopintegrationmomentaaroundthecorresponding
dominantdirection: k =x p+(k ·p)n+k withinthecertainlightconebasisformedbythevectors
i i i T
pandn(inourcase, n∗ andn); (ii)the replacement: d4k =⇒d4k dxδ(x −k ·n)thatintroduces
i i i i i
thefractionswiththeappropriatedspectralproperties;(iii)thedecompositionofthecorresponding
propagatorproductsaroundthedominantdirection:
(cid:12)
H(k)=H(xp)+∂H(k)(cid:12)(cid:12) kT +...;
∂k (cid:12) ρ
ρ k=xp
(iv)theuseofthecollinearWardidentity,ifitrequestsbytheneededapproximation:
∂H(k)
=H (k,k);
ρ
∂k
ρ
3
Gaugeinvarianceandgluonpolesfordirectphotonproduction I.V.Anikin
(v) the Fierz decomposition for ψ (z)ψ¯ (0) in the corresponding space up to the needed projec-
α β
tions. Noticethat,forourpurposes,itisenoughtobelimitedbythefirstorderofdecompositionin
thethirditem. Asaresultofthisprocedure,weshouldreachthefactorizedformfortheconsidered
subject:
Hadron tensor={Hard part (pQCD)}⊗{Soft part (npQCD)}. (3.1)
Usually, both the hard and soft parts, see (3.1), are independent of each other, UV- and IR-
renormalizable and, finally, various parton distributions, parametrizing the soft part, have to man-
ifest the universality property. However, the hard and soft parts of the DPP hadron tensor are not
fullyindependent eachother[1,19]. Actually, theDY hadrontensorhas formally factorizedwith
themathematicalconvolutionandthetwist-3functionBV(x ,x )satisfiesstilltheuniversalitycon-
1 2
dition. IncontrasttotheDY-process,theDPPtensorwillincludethefunctionsBV(x ,x )thatwill
1 2
notmanifesttheuniversality.
4. QCDgaugeinvarianceofthehadrontensor
We now dwell on the QCD gauge invariance of the hadron tensor for the direct photon pro-
duction (DPP). First of all, let us remind that having used the contour gauge conception [19, 23],
onecancheckthattherepresentation
T(x ,x )
BV(x ,x )= 1 2 (4.1)
+ 1 2 x −x +iε
1 2
belongstothegaugedefinedby[x,−∞]=1,whiletherepresentation
T(x ,x )
BV(x ,x )= 1 2 (4.2)
− 1 2 x −x −iε
1 2
correspondstothegaugethatdefinedby[+∞,x]=1. Inboth(4.1)and(4.2),thefunctionT(x ,x )
1 2
relatedtothefollowingprametrization:
(cid:104)p ,ST|ψ¯(λ n˜)γ+n˜ Gνα(λ n˜)ψ(0)|ST,p (cid:105)=
1 1 ν T 2 1
(cid:90)
εα+ST−(p p ) dx dx eix1λ1+i(x2−x1)λ2T(x ,x ). (4.3)
1 2 1 2 1 2
Roughly speaking, it resembles the case where two different vectors have the same projection on
thecertaindirection. Inthissense,theusualaxilgaugeA+=0canbeunderstoodasa“projection"
whichcorrespondstotwodifferent“vectors"representedbytwodifferentcontourgauges.
Further, to check this invariance, we have to consider four typical diagrams H1, H5, D1 and
H9,depictedinFig.2,thatcorrespondtothecertainξ-process(see,[27]). Noticethat,fortheQCD
gauge invariance, we have to assume that all charged particles are on its mass-shells. That is, we
willdealwithonlythephysicalgluons.
To write down the Ward identity, we need to replace the gluon transverse polarization εT on
α
thegluonlongitudinalmomentum(cid:96)L inthequark-gluoncorrelator:
α
(cid:90)
Φ¯[γ+],ρ(k ,(cid:96))=− (d4η )e−ik1η1ερ(cid:104)p ,ST|ψ¯(0)γ+ψ(η )a+((cid:96))|ST,p (cid:105)ε=T→⇒(cid:96)L
⊥ 1 1 T 1 1 1
(cid:90)
− (d4η )e−ik1η1(cid:96)ρ(cid:104)p ,ST|ψ¯(0)γ+ψ(η )a+((cid:96))|ST,p (cid:105). (4.4)
1 L 1 1 1
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Gaugeinvarianceandgluonpolesfordirectphotonproduction I.V.Anikin
Here,a+((cid:96))standsforthegluoncreationoperatorandthesummationovertheintermediatestates
are not shown explicitly. Notice that the parametrization of this correlator through BV-function
leaveswithnoanychangesintheform.
Consider now the contribution of the H1-diagram, depicted in Fig.2, to the hadron tensor.
Beforegoingfurther,itisinstructivetobeginwiththegluonloopintegrationcorrespondingtothe
mentioneddiagram,wehave
(cid:90)
(d4(cid:96))S((cid:96)+k+q)(cid:96)ˆ (cid:104)...a+((cid:96))...(cid:105), (4.5)
L
where we do not write explicitly the irrelevant, at the moment, operators (cf. (4.4)). After factor-
ization,weobtain
(cid:90) (cid:90)
dx (d4(cid:96))δ(x −x −(cid:96)n)S((cid:96)+k+q)(cid:96)ˆ (cid:104)...a+((cid:96))...(cid:105)=
2 2 1 L
(cid:90) (cid:90)
dx S(x p +yp )(x −x )pˆ (d4(cid:96))δ(x −x −(cid:96)n)(cid:104)...a+((cid:96))...(cid:105), (4.6)
2 2 1 2 2 1 1 2 1
where we decomposed the hard part around the dominant direction and used (cid:96) = (x −x )p ,
L 2 1 1
which is actually dictated by the γ-structure, together with the momentum conservation. With
these,wehavethefollowingexpression:
(cid:90)
W(diag.H1)=−2 dx dyδ(4)(x p +yp −k−q)Fg(y)C ×
1 1 1 2 2
γ+ (cid:90) (x −x )γ+γ− γ+
v¯(k)εˆ γ− dx 2 1 εˆ∗v(k)×
2
2x p +iε 2x +iε 2x p +iε
1 1 2 1 1
(cid:110) (cid:90) (cid:90) (cid:111)
(−) (dλ )e−ix1λ1(cid:104)p ,ST|ψ¯(0)γ+ψ(λ n) (d4(cid:96))δ(x −x −(cid:96)n)a+((cid:96))|ST,p (cid:105) . (4.7)
1 1 1 2 1 1
Wecanhereusetheparametrizationintheformof
(cid:90) (cid:90)
(−) (dλ )e−ix1λ1(cid:104)p ,ST|ψ¯(0)γ+ψ(λ n) (d4(cid:96))δ(x −x −(cid:96)n)a+((cid:96))|ST,p (cid:105)
1 1 1 2 1 1
=BV(x ,x ). (4.8)
1 2
It is seen, however, that in any case this diagram does not contribute to the Ward identity after
calculationoftheimaginarypartowingtothefactor(x −x )inthenumeratorof(4.7).
2 1
Further,calculationoftheH5-diagram,presentedinFig.2,givesus
(cid:90)
W(diag.H5)= dx dyδ(4)(x p +yp −k−q)Fg(y)C ×
1 1 1 2 2
γ+ (cid:90) γ+γ−γ+
v¯(k)εˆ γ−εˆ∗ dx v(k)BV(x ,x ), (4.9)
2 1 2
2x p +iε 2x p +iε
1 1 2 1
whilethecontributionoftheD1-diagraminFig.2takestheform
(cid:90)
W(diag.D1)=− dx dyδ(4)(x p +yp −k−q)Fg(y)C ×
1 1 1 2 1
γ+ γ+ (cid:90)
v¯(k)εˆ γ−γ+γ− εˆ∗v(k) dx BV(x ,x ). (4.10)
2 1 2
2x p +iε 2x p +iε
1 1 1 1
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Gaugeinvarianceandgluonpolesfordirectphotonproduction I.V.Anikin
And,finally,thecontributionoftheH9-diagramwiththethree-gluonvertex,seeFig.2,reads
(cid:90)
W(diag.H9)=−4i dx dyδ(4)(x p +yp −k−q)Fg(y)C ×
1 1 1 2 3
γ+ γ+ (cid:90) x −x
v¯(k)εˆ γ− εˆ∗v(k) dx 2 1 BV(x ,x ). (4.11)
2 1 2
2x p +iε 2x p +iε 2(x −x )+iε
1 1 1 1 2 1
We now turn to the contour gauge. Based on the discussions in [19], even at glance, we are able
to anticipate the corresponding prescriptions for BV-functions in (4.9)–(4.11). Indeed, the H5-
diagram in Fig.2 corresponds to the final state interaction and, therefore, the function BV should
−
appear here, while the D1- and H9-diagrams in Fig.2 – to the initial state interaction which leads
to the function BV. Performing the explicit calculations (as we did for the DY-process [1]), we
+
canobtainthesameconclusionbyrestoringtheWilsonlinesinthequark-gluoncorrelatorsofthe
mentioneddiagrams. Thatis,theWilsonline[+∞−,z−]willenterinthehadrontensorrepresented
bytheH5-diagraminFig.2;theWilsonline[z−,−∞−]willstandinthehadrontensorrepresented
bytheD1-andH9-diagramsinFig.2.
Sothat,wesumallcontributionsandgetthefollowingfinalexpression:
C (cid:90) BV(x ,x ) C (cid:90)
∑W(diag.N) = 2 γ+γ−γ+γ−γ+ dx − 1 2 + 1 γ+γ−γ+γ−γ+ dx BV(x ,x )+
8x 2 x 8x2 2 + 1 2
N 1 2 1
iC (cid:90) (x −x )BV(x ,x )
3γ+γ−γ+ dx 2 1 + 1 2 (4.12)
4x2 2 x −x +iε
1 2 1
wherethefunctionBV aregivenby(4.1)and(4.2).
±
Wenowcalculatetheimaginarypartand,ultimately,derivetheQCDWardidentityintheform
C −C −iC =−[ta,tb]tbta+ifabctctbta≡0. (4.13)
2 1 3
We want to stress that the identity (4.13) takes place provided only the presence of the different
complexprescriptionsingluonicpolesdictatedbythefinalorinitialstateinteractions:
FSI ⇒ 1 ⇒gauge [+∞−,z−]=1⇒ T(x1,x2) (cid:41)
(cid:96)++iε x1−x2−iε
⇒QCD GI. (4.14)
ISI ⇒ 1 ⇒gauge [z−,−∞−]=1⇒ T(x1,x2)
−(cid:96)++iε x1−x2+iε
We emphasize the principle differences between the considered case and the proof of the QCD
gaugeinvariancefortheperturbativeComptonscatteringamplitudewiththephysicalgluonsinthe
initial and final states. The latter does not need any external condition, like the presence of gluon
poles.
Thus, the situation which we discuss is again absolutely similar to that one which was de-
scribed in [26] for the dijet production. From (4.14), it is seen that the different diagrams corre-
spondtothedifferentcontourgaugesand,consequently,todifferentthefunctionsBV thatparametrize
±
thehadronicmatrixelementformingthesoftpart. Inthiscontext,wealsohaveasoftbreakingof
factorization because, first, it spoils the universality principle and, second, the gluonic pole pre-
scriptionsinthesoftparthavebeentracedtothecausalprescriptionsinthehardpart. Besides,the
possiblereasonsforthecollinearfactorizationbreakinghasbeenpresentein[19].
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Gaugeinvarianceandgluonpolesfordirectphotonproduction I.V.Anikin
p1 pp11 pp11 p1
p2 pp22 pp22 p2
H1 H5 H9
p1 p1
p2 p2
D1
Figure2: ThetypicalFeynmandiagramstochecktheQCDgaugeinvariance.
5. Conclusions
We explore the QCD gauge invariance of the hadron tensor for the direct photon production
intwohadroncollisionwhereoneofhadronsistransverselypolarized. Wearguetheeffectswhich
lead to the soft breaking of factorization by inspection of the QCD gauge invariance. We demon-
strate that the initial or final state interactions in diagrams define the different prescriptions in the
gluonic poles. Moreover, the different prescriptions are needed to ensure the QCD gauge invari-
ance. This situation can be treated as a soft breaking of the universality condition resulting in
factorizationbreaking.
Acknowledgements
We would like to thank I.O. Cherednikov, A.V. Efremov, D. Ivanov and L. Szymanowski for
usefuldiscussionsandcorrespondence. ThisworkispartlysupportedbytheHLprogram.
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