TMD Evolution at Moderate Hard Scales

0.85 MB·

by  Ted C. Rogers

TMD Evolution at Moderate Hard Scales
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2 Ted C. Rogers
∗

n DepartmentofPhysics,OldDominionUniversity,Norfolk,VA23529,USA
a
andTheoryCenter,JeffersonLab,12000JeffersonAvenue,NewportNews,VA23606,USA
J
E-mail: [email protected]
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1
John Collins

] 104DaveyLab.,PennStateUniversity,UniversityParkPA16802,USA
h
E-mail: [email protected]
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-
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e We summarize some of our recent work on non-perturbative transverse momentum dependent
h
[ (TMD)evolution,emphasizingaspectsthatarenecessaryfordealingwithmoderatelylowscale

processeslikesemi-inclusivedeepinelasticscattering.
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v PoS.cls, January 11, 2016, JLAB-THY-16-2196, DOE/OR/23177-3644
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QCDEvolution2015-QCDEV2015-
26-30May2015
JeffersonLab(JLAB),NewportNewsVirginia,USA
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
TMDEvolutionatModerateHardScales TedC.Rogers
1. TMDfactorizationandnon-perturbativeevolution
ThepurposeofthistalkistosummarizeresultsrecentlypresentedinRef.[1]. Wewilldiscuss
the Collins-Soper-Sterman (CSS) form of TMD factorization in the updated version presented in
Ref. [2]. (See Ref. [3] for a general overview and for references.) For these proceedings, the
relevantaspectsoftheTMDfactorizationtheoremsarethefollowing:
TheunpolarizedcrosssectionforaprocesslikeDrell-Yanscatteringisexpressibleas
•
dσ
d4qdΩ
2 dσˆ (Q,µ Q;α (Q))
= ∑ j¯ → s d2b eiqTb F˜ (x ,b;Q2,Q)F˜ (x ,b;Q2,Q)
s dΩ · j/A A ¯/B B
j (cid:90)
+largeq “Y-term”correction. (1.1)
T
wheredσˆ /dΩisahardpartoniccrosssectionandF˜(x,b;Q2,Q)isaTMDpartondistribu-
j¯
tionfunction(PDFs)incoordinatespaceevaluatedwithahardscaleQ.
Collins-Soper(CS)evolutionappliedtoanindividualTMDPDFleadsto
•
∂
lnF˜ (x ,b ;Q2,Q)=K˜(b ;Q)+ b IndependentTerms (1.2)
∂lnQ j/A A T T T
The“b IndependentTerms”onlyaffectthenormalizationofF˜ butnotitsshape.
T
The kernel K˜(b ;Q) is is strongly universal. At small b its b -dependence is perturba-
T T T
•
tively calculable with 1/b acting as a hard scale. At large b its b -dependence is non-
T T T
perturbative.
Forallb ,K˜(b ;Q)obeystherenormalizationgroup(RG)equation:
T T
•
d
K˜(b ;µ)= γ (α (µ)). (1.3)
T K s
dlnµ −
Atsmallb ,onehopestoexploitperturbationtheorywith1/b asahardscaletocalculateK˜(b ;Q)
T T T
whileatlargeb anon-perturbativeparametrizationisneeded. Inthenon-perturbativeregion,one
T
hopestoexploitthestronguniversalityofK˜(b ;Q)tomakepredictions. Oneneedsaprescription
T
todemarcatewhatconstituteslargeandsmallb . Tosmoothlyinterpolatebetweenthetworegions,
T
oneimposesagentlecutoffonlargeb . Acommonchoiceofcutofffunctionis
T
b
T
b (b )= . (1.4)
T
∗ 1+b2/b2
T max
(cid:113)
Then an RG scale defined as µ C /b approaches C /b at small b and C /b at large
b 1 1 T T 1 max
b . We can separate K˜(b ;Q) int∗o≡a large∗b part and a small b part by adding and subtracting
T T T T
K˜(b ;Q)inEq.(1.2):
∗
∂
lnF˜ (x ,b ;Q2,Q)=K˜(b ;Q)+ K˜(b ;Q) K˜(b ;Q) + b IndependentTerms. (1.5)
∂lnQ j/A A T ∗ T − ∗ T
(cid:2) (cid:3)
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TMDEvolutionatModerateHardScales TedC.Rogers
Theg (b ;b )functionisdefinedasthetermK˜(b ;Q) K˜(b ;Q),sothat
K T max T
− ∗
∂
lnF˜ (x ,b ;Q2,Q)=K˜(b ;Q) g (b ;b )+ b IndependentTerms. (1.6)
∂lnQ j/A A T ∗ − K T max T
Bydefinition,therightsideofEq.(1.6)isexactlyindependentofb . FromEq.(1.3),g (b ;b )
max K T max
is also exactly independent of Q. The Q dependence in each of the terms in the definition of
g (b ;b ) cancels. We can apply Eq. (1.3) to exploit RG improvement in the calculation of
K T max
K˜(b ;Q):
∗ Q dµ
K˜(b ;Q)=K˜(b ;µ ) (cid:48)γ (α (µ )). (1.7)
b K s (cid:48)
∗ ∗ ∗ −(cid:90)µb µ(cid:48)
∗
So,theevolutionoftheshapeofF˜ (x ,b ;Q2,Q)isisgivenby
j/A A T
∂
lnF˜ (x ,b ;Q2,Q)
∂lnQ j/A A T
Q dµ
=K˜(b ;µ ) (cid:48)γ (α (µ )) g (b ;b )+ b IndependentTerms. (1.8)
b K s (cid:48) K T max T
∗ ∗ −(cid:90)µb µ(cid:48) −
∗
The partial derivative symbol means x is to be held fixed. The g (b ;b ) function inherits the
A K T max
universality properties of K˜(b ;µ). In particular, it is related to the vacuum expectation value of
T
a relatively simple Wilson loop. It is independent of any details of the process and is even the
same if the PDF F˜ (x ,b ;Q2,Q) is replaced with a fragmentation function. Thus we say that
j/A A T
g (b ;b ) is “strongly” universal; see the graphic in Fig. 1. The g (b ;b ) function is often
K T max K T max
called the “non-perturbative” part of the evolution since it can contain non-perturbative elements.
This is a slight misnomer, however, since g (b ;b ) can contain perturbative contributions as
K T max
well. Indeed,atverysmallb itisentirelyperturbativelycalculable,thoughsuppressedbypowers
T
ofb /b ,accordingtoitsdefinitioninEq.(1.5).
T max
2. Largeb behavior
T
Acommonchoicefornon-perturbativeparametrizationsofg (b ;b )isapower-lawform.
K T max
These tend to yield reasonable success in fits that involve at least moderately high scales Q [4].
However, extrapolations of those fits to lower values of Q (such as those corresponding to many
currentSIDISexperiments)appeartoappeartoproduceevolutionthatisfartoorapid[5,6]. Inthis
talk, we carefully examine the underlying physics issues surrounding non-perturbative evolution
and, on the basis of those considerations, we will propose a form for g (b ;b ) that accommo-
K T max
datesbothlargeandsmallQbehavior.
Wewillfirstwritedownourproposedansatzforg (b ;b )andthenspendtheremainderof
K T max
thetalkdiscussingitsjustifications. Ourproposalis
C α (µ )b2
g (b ;b )=g (b ) 1 exp F s b T , (2.1)
K T max 0 max − −πg (b )∗b2
(cid:18) (cid:20) 0 max max(cid:21)(cid:19)
where
g (b )=g (b )+2CF C1/bmax dµ(cid:48)α (µ ). (2.2)
0 max 0 max,0 s (cid:48)
π (cid:90)C1/bmax,0 µ(cid:48)
3
14
kˆ = xP+,0,0
T
ˆ
l =�0,q�,0T �
� �
k˜ = xP+,k ,k
� T
˜l =�l+,q�,kT �
� �
dk+ dk d2k (244)
� T
Z Z Z
sˆ (245)
sˆ Q2(1 x/⇠) Q2(1 z)
k2 = = � = � (246)
T,max 4 4(x/⇠) 4z
Q2(1 z)
sˆ= (q+kˆ)2 = � (247)
z
kT2,max d�ˆ
�ˆ = (248)
dk2
Z T
1 dk2 (249)
T
Z0
d�
d4qd⌦
2 d�ˆ (Q,µ,↵ (µ))
= j|¯ s d2bT eiqT·bT F˜j/A(xA,bT;⇣A,µ) F˜|¯/B(xB,bT;Q4/⇣A,µ) 16
s d⌦
j Z
X
+polarization terms+high-q term (Y)+power-suppressed. (250)
T
(276)
d�ˆ (Q,µ,↵ (µ)) s
W˜ (b ;Q) Q2 j|¯ s F˜ (x ,b ;⇣ ,µ)F˜ (x ,b ;Q2/⇣ ,µ) (251)
T j j/A A T A |¯/B B T A
⌘ d⌦
d� qT⌧Q 2 dF2˜bfT/Pei(qbT·TbT,Wx˜;(µbT;⇣;Q1)) = (252) (277)
d4qd⌦ ' sQ2
Z
@lnW˜ (bT,Q,xA,xB) = K˜(b ;⇣µ1)+=bx2MInpd2eep2e(ynPd�enyts)Terms (253) (278)
T T
@lnQ2
d x 1.0 (279)
˜
K(b ;µ) = � (↵ (µ)) (254)
TMDEvolutionatModeratedHlanrdµScalesT � K s! TedC.Rogers
0 WL Q[a,daµ] 0 (280)
˜ ˜ C 0
K(bT;Q;↵s(Q)) = K(bT;µb ;↵s(hµb|))+ �|Ki(↵s(µ0)) (255)
µ
⇤ ⇤ Zµb 0
⇤
l+l-­‐

 t¯o

K˜(b ;µ) ↵ (µF).Tln.(µPbbac) k+-­‐t(o0-­‐b,awc+k�

 O,0(t↵)(µ)nln(µb )(m0,)0,0t) P (256) (281)
T s h T| s T | i
⇠ ···
hadrons
  Z/W
 produc:on

Drell-­‐Yan

µ = C /b µ = C /b (257)

  b 1 T ¯ b 1
F.T. P (0,w⇤�,wt) ⇤ (0,0,0t) P (282)
h | | i
˜ ˜ ˜ ˜
K(b ;µ ;↵ (µ )) = K(b ;µ ;↵ (µ ))+K(b ;µ ;↵ (µ )) K(b ;µ ;↵ (µ )) (258)
T 0 s 0 0 s 0 T 0 s 0 0 s 0
SIDIS
  ⇤ � ⇤
Collins

  Q
gK(bT;bmax)ln
  (259) (283)
� Q
0
................... ✓ ◆
...................
Boer-­‐

PDFs

 Mulders

Fragmenta:on
  Acknowledgments
Sivers

Func:ons
  Unpolarized

  Cross
 Sec:ons

This work was supported by...

Figure 1: Strong universality of the non-perturbative evolution parametrized by g (b ;b ). The
K T max
g (b ;b )ln(Q/Q )combinationappearsexponentiatedintheevolvedcrosssectionexpression.
K T max 0
−
Theonlyparameterofthemodelisg (b )anditvarieswithb accordingtoEq.(2.2). b
0 max max max,0
isaboundaryvalueforg relativetowhichothervaluesaredetermined.
0
First,notethattheasmallb /b expansionofEq.(2.1)gives
T max
C b2 b4C2α (µ )2
g (b ;b )= F T α (µ )+O T F s b , (2.3)
K T max π b2 s b b4 π2g (b∗ )
max ∗ (cid:18) max 0 max (cid:19)
whileanexpansionoftheexactdefinitionof g (b ;b )inEq.(1.5)is
K T max
−
g (b ;b )= K˜(b ;µ ;α (µ ))+K˜(b ;µ ;α (µ ))
K T max T b s b b s b
− ∗ ∗ ∗ ∗ ∗
C b2 b4
= F T α (µ )+O T α (µ )2 (2.4)
π b2 s b π2b4 s b
max ∗ (cid:18) max ∗ (cid:19)
So,theexactdefinitionandEq.(2.1)matchinthesmallb limit.
T
3. Conditionsong (b ;b )
K T max
Ourdescriptionofthelargeb limitofcorrelationfunctionslikeF˜(x ,b ;Q2,Q)ismotivated
T A T
bythegeneralobservationthattheanalyticpropertiesofcorrelationfunctionsimplyanexponential
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TMDEvolutionatModerateHardScales TedC.Rogers
coordinatedependence,withapossiblepower-lawfall-off,forthelargeb limit. Thatis,neglecting
T
perturbativecontributions,
1
F˜(xA,bT;Q2,Q)bT∼→∞ b αe−mbT, (3.1)
T
with m and α independent of Q. See, for example, Ref. [7]. Therefore, from Eq. (1.2), K˜(b ;Q)
T
mustapproachab -independentconstantatlargeb .
T T
Thesetofrequirementsong (b ;b )is
K T max
1. K˜(bT;µb )bT=→0K˜(bT;C1/bT)iscalculableentirelyinperturbationtheorywithC1/bTplaying
∗
theroleofahardscale.
2. K˜(b ;Q)approachesaconstantatb /b ∞. TheconstantcanbeQ-dependent, butthe
T T max
→
Q-dependencecanbecalculatedperturbativelyforallb fromEq.(1.3).
T
3. Because of item 2, g (b ;b ) must approach a constant at large b , but the constant de-
K T max T
pendsonb .
max
4. At small b , g (b ;b ) is a power series in (b /b )2 with perturbatively calculable
T K T max T max
coefficients,asinEqs.(2.3,2.4).
5. Bydefinition,therightsideofEq.(1.8)isindependentofb andthisshouldbepreserved
max
asmuchaspossibleinthefunctionalformthatparametrizesg (b ;b ). Forsmallb ,this
K T max T
means
d d
asy g (b ;b ) = asy g (b ;b ) (3.2)
K T max K T max
db db
bT bmax max (cid:12)parametrized bT bmax max (cid:12)truncatedPT
(cid:28) (cid:12) (cid:28) (cid:12)
(cid:12) (cid:12)
where“parametrized”referstoa(cid:12) specificmodelofg (b ;b )while“t(cid:12)runcatedPT”refers
K T max
to a truncated perturbative expansion. Eqs. (2.3,2.4) satisfy this requirement through order
α (µ ).
s b
∗
6. Atlargeb ,b -independenceoftheexactK˜(b ,µ)impliesthat,toausefulapproximation,
T max T
d dK˜(b ;C /b )
max 1 max
g (b =∞;b )= γ (α (C /b )) , (3.3)
K T max K s 1 max
dlnb dlnb −
max (cid:20) max (cid:21)truncatedPT
as obtained from Eq. (1.7) and the definition of g . Equation (2.2) ensures that Eq. (2.1)
K
satisfies Eq. (1.8) so long as everything is calculated only to order α (µ ). Enforcing both
s b
∗
Eq.(3.2)andEq.(3.3)simultaneouslymeansg (b ;b )willproduceab independent
K T max max
contribution to K˜(b ;Q) for all b except perhaps for an intermediate region at the border
T T
between perturbative and non-perturbative b -dependence. The residual b dependence
T max
therecanbereducedbycalculatinghigherordersandrefiningknowledgeofnon-perturbative
behavior.
For a much more detailed discussion of these considerations, see Sect. VII of Ref. [1]. Equa-
tion(2.1)isoneofthesimplestmodelsthatsatisfiesall6ofthesepropertiessimultaneously.
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TMDEvolutionatModerateHardScales TedC.Rogers
4. Conclusion
In Sect. 3 we enumerated properties that a model of g (b ;b ) needs tp ensure basic con-
K T max
sistencyinacalculationK˜(b ;Q). AsimpleparametrizationwasproposedinSect.2.
T
Note that a quadratic (b /b )2 dependence at small b emerges naturally from (2.1), but
T max T
with a perturbatively calculable coefficient. Furthermore, the dependence is not exactly quadratic
becausethecoefficientscontainlogarithmicb dependencethroughα (µ ).
T s b
∗
In a process dominated by very large b , Sect. 3 and Eq. (2.1) predict an especially simple
T
evolution for the low-Q cross section. Namely, the cross section scales as (Q/Q )a where a is
0
combinationofg (∞,b )andperturbativelycalculablequantities. (SeeEq.(85,86)ofRef.[1].)
K max
Futurephenomenologicalworkshouldincludeeffortstoconstraing . Becauseofitsstrongly
0
universalnature,thisoffersarelativelysimplewaytotestTMDfactorization.
Acknowledgments
ThisworkwassupportedbyDOEcontractNo.DE-AC05-06OR23177,underwhichJefferson
ScienceAssociates,LLCoperatesJeffersonLab.,andbyDOEgrantNo.DE-SC0013699.
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