Finding patterns within perturbative approximation in QCD and indirect relations

0.17 MB·English

by  M. R. Khellat

Finding patterns within perturbative approximation in QCD and indirect relations
M. R. Khellat∗
Physics Department, Yazd University, 89195-741, Yazd, Iran
(Dated: January 17, 2011)
WemakeanextensiveuseofBLMapproachtostudydetailsofpredictinghigherordercorrections
based on the approach. This way we are able to test two procedures to improve the prediction
process. Besidethemain lineofthetwoproceduresitisfoundoutthatoverallnormalization could
changeBLMpatternseffectively. Finally wetrytofindoutwhetheraBLMpatternissufficientfor
a prediction or not, and how oneshould use such a pattern.
PACSnumbers: 11.15.Bt12.38.Bx11.90.+t
1
1
0 I. INTRODUCTION Thiswayweencounteranewfaceoftheobservable. This
2 mathematical adaptation can be interpreted as isolating

n Optimizationprocedures providea goodframeworkin parts of the observable. These isolated parts carry the
a investigatingthebehaviorofperturbativedescriptionsof physical considerations which supports the optimization
J gauge field theories’ observables. Perturbative analysis procedure as parts of them. The set of isolated parts
4 of these theories encounters the problem of scheme-scale can be called a pattern. However this is not the only
1 ambiguity. An appropriate approach to this problem way to find patterns. Regarding the approach extension

should give warnings in cases where it should not be ap- andimprovement,weintroducebiasedBLMapproaches.
]
h plied, orweshouldhavea mechanismto checkthe valid- ApparentlyanormalBLMapproachcannotcorrectlyan-
p ity. BLM approach [1] contains such a mechanism, it is alyzegluon-gluoninteractionsin leading order. Here the
p- not applicable in analyzing processes containing gluon- suggestion of finding BLM generated parts(the isolated
e gluon interactions in leading order since we should not partsgeneratedthroughBLMpredictioncomparison)in-
h absorballtheflavordependentpartintotherunningcou- side the observable and using the new obtained parame-
[ pling constant in these cases. If we use the approach to tersinsteadoftheoriginalonesconstitutesanimmediate

3 resolveschemeambiguityandfindphysicalschemes,such biasing. Also we introduce a second-class biasing where
v processes will automatically be eliminated. Strange per- the leading-order is totally dismissed from prediction.
1 turbative series would be the outcome of using them as From a more illuminative point of view, we check the
3 the physical scheme. The mathematical form of pertur- potential of the approach in revealing a pattern for the
5 bativeseriessuggestsakindofsymmetryandrelationbe- observable. InthecaseofBLMthishasbeenpreferredto
2
tween observables but without considering optimization finding a way to separate gluon-gluon interactions from
.
2 processes the real hidden relation can not be revealed. the vacuum polarizations. If the approach can be ad-
1 BLM approach is particularly able to reveal commensu- justed to find a satisfying pattern, it has jumped over
0
rate scale relations [2] between observables. This flexi- the obstacle.
1
bility of BLM approach can provide us with very simple Following the route of finding patterns will lead us
:
v relations such as the generalized Crewther relation [3] through a pile of equations to a simple sufficient con-
Xi which can be used to test some serious aspects of QFT dition for the relations that connects observables. The
such as the violation of conformal symmetry due to the condition is a sufficient one if someone wants to get new
r
a renormalizationprocedure. results from prediction through a second observable.
Onecanuseoptimizationprocedurestopredicthigher- For the convenience of the reader: firstly in this way
ordercoefficients. Clearlyanykindofpredictionstrongly of finding patterns and biasing we deal with systems of
assumes existence of patterns. A pattern at least must equations that are built upon comparisons. These com-
not have mathematical defects. However the quality of parisonsmayaimdifferenttypesofparametersinthepat-
a proposed pattern is usually determined through the tern such as the perturbative series coefficients or solely
comparison of the predicted and exact coefficients. For the local symmetry parameters. Secondly we make no-
a flexible approach this comparison could be a guideline ticesanywhereonemayfindhimselfinastatetoperform
in improving the approach. As a matter of fact what aselectionofequationsorparameterssuchaswhendeal-
should be followed here rather than improving an ap- ing with unequal number of equations and parameters.
proach is finding patterns. Fortunately we do not have In the beginning of Sec.II we will discuss primary as-
many choices for the first step. The first step towards pects of prediction. Deviation pattern approachis intro-
theseaimsistomakethepredictionrebuildexactvalues. ducedinSec.IIAaccompaniedby itsdetailsandusesfor
Higgs boson decay widths, Bjorken sum rule and Adler
function and also some complementary discussions on
prediction. In this section we also consider patterns de-
∗ [email protected] veloped from the combination of different patterns. As
2
we go further we will try take advantage of a second ob- of the representation is T for the fundamental repre-
F
servable for finding patterns in Sec.IIB and introduce sentation. Obviously setting N , the number of gener-
A
indirect relations. ators of the group, C , T , and dabcd, the higher order
F F F
group invariants, to 1 and the parameters of the adjoint
representation to 0 will separate the U(1) factor of the
II. THE PROCEDURES group. The dynamics of the quenched approximation of
the non-Abelian part is revealed in the limit n = 0 for
f
all physical energies while the conformal invariant part-
We consider a RG-invariant perturbative expansion
R(Q2)= r +r a (Q2)+ ∞ r a i(Q2)wherea = αs ner of the symmetry group sits where the β-function is
0 1 s i=0 i s s π 0.
(αs is the RG-invariant effPective coupling constant) and
r = i−1 r nk for any effective charge or MS-like
i k=0 ik f
schemeP. Predicting any coefficient ri using BLM ap- A. Deviation
proachresultsinaseriesofnumber offlavorsn forthat
f
orderthatdependsonallthelowerordercoefficients. For
instance the predicted third order coefficient is If BLM prediction pattern is some good pattern for
the perturbative series, one can rewrite the whole se-
rpre = 121CA2 r221 17CA2r 11CA r20r21 ries in terms of just r1 and r2 coefficients. The quality
3 − 16 T2 r − 8 T 21 − 2 T r of the pattern will depend on these parameters beside
F 1 F F 1
the BLM approach itself or truthfully the mathematical
r r 5 3 r2
+(2 20 21 + r C + r C )n + 21n2. (1) mechanismemployedfor it. Anywayif BLMapproachis
r 4 21 A 4 21 F f r f
1 1 able to discover the pattern, it will not be so surprising
if the original r and r do not be the ones that develop
Here we have used the 4-dimensional β-function of the 1 2
the pattern. A straightforward step would be to bor-
massless MS-like schemes. The nth order predicted co-
efficient rpre = n−1rprenk appears when re-expanding row the r1 and r2 that form the real r3 since it is the
n k=0 nk f
most important part of the pattern. Yet the procedure
the BLM scale-fiPxed (n 1)th order series
− wouldbeflexibleenoughtobeabletohandleexceptional
R=r +r a(Q∗2)+r¯ a2(Q∗2)+...+r¯ an−1(Q∗2). but important cases such as the ones with large higher-
0 1 2 n−1
order corrections. We could discuss the situation also as
Q∗ is to absorb all vacuum polarization. Re-expanding if the weights of r1 and r2 parts should be other values
the above series happens through the nth order running inEq.(1)orothercombinationsofweightsoflower-order
of the couplant: coefficients are appropriate.
In general a BLM pattern contains several coupled
a(Q∗2)=a(Q2) a2(Q2)β ℓ+a3(Q2)(β2ℓ2 β ℓ) equations corresponding to all we know about the per-
− 0 0 − 1
5 turbative series. These equations are acquired by com-
+a4(Q2)( β3ℓ3+ β β ℓ2 β ℓ)+...+ (an+1),
− 0 2 0 1 − 2 O paring the predicted coefficient and the exact ones. For
instance when one knows r in addition to r and wants
4 3
whereℓ=ln(Q∗2/Q2)anditmayoccurinmultiplesteps toadaptthepredictionpatternsothatallthedetailsare
with appropriate ℓs for a multi-scale versionof the BLM involvedinit,hewillnoticethatrpredependsonallthe6
4
scale-fixed series. We will discuss more details of BLM partsoflowerordercoefficients r ,r ,r ,r ,r ,r .
1 20 21 30 31 32
extensions later. But the 4th order comparison(r{pre = r ) yields 4 equa}-
4 4
One more preliminary review on overall normaliza- tions. Consideringthe3rdordercomparison,weseethat
tion and Casimir operators is needed before going on to 7 equations are controlled by just 6 variables. This in-
the procedures. LocalgaugeinvarianceconstructsYang- dicates that one of these 7 parts is generated automat-
Mills theories i.e., non-Abelian gauge theories. Such lo- ically through the adaptation of the other parts. The
cal symmetries suites well to compact semi-simple Lie more information, the more automatic generation. So
algebras. For such an algebra one can find the overall here one is forced to do a selection of equations consid-
normalization [4–6] that can determine the structure of ering the importance of the corresponding comparison.
the algebra and its representations. The tensor prod- For instance one may find out that rpre plays the least
43
uct of generators of the color SU(N)s of two particles, importantrole in the deviationof rpre from r , the com-
4 4
TATB,givesthetasteofcolorinteractionbetweenthem. parison rpre = r will be easily removed from the list
a a 43 43
This operator is surprisingly an invariant of the tensor of equations. The same works for other selections of pa-
product space. The Casimir operator T2, where T s are rameters too. Deviation Pattern procedure obtains new
a a
generators of the product space, related to this inter- parameters through a selection of comparisons and then
action is a number depending on the dimension of the replaces the original parameters with the new ones to
representation. C and C are quadratic Casimir op- build the higher-order coefficients.
A F
erators of the adjoint and fundamental representations It is possible to involve Adler transformation into the
transforming gauge and fermion fields. The representa- aboveprocedurebyperformingthepredictionorcompar-
tion’sgeneratorsscalefactori.e.,thetracenormalization ison for the corresponding Adler function. Some notes
3
regradingperturbativeaspects ofAdler functionwill fol- tion. Considering M >>2m results in
H q
low. The real function R(s) where s denotes the center-
of-mass energy squared is related to the Adler function Γ(H qq¯)= 3GF m2R˜(s)
D(Q2) through → 4√2πM q
H
D(Q2)=Q2 ∞ ds R(s). for the decay width, where the term to the left of R˜
Z0 (s+Q2)2 is ΓBorn(H qq¯). The absorptive part of the corre-
→
sponding two pointcorrelatorR˜ generatesthe QCD cor-
The reverse transformation rebuilds R(s) as follows:
rections. Gluonic decay width takes the following form
i s+iǫ dz based on the fact that top quarks plays the main role in
R(s)= D ( z).
pt coupling of gluons to Higgs boson,
2π Z z −
s−iǫ
One may use the RG-invariant perturbative approxima- √2G
tionoftheAdlerD-functionDpt(Q2)= ndnαns(Q2)in Γ(H →gg)= MHFC12ImΠGG(q2),
the reverse transformation. This will gePnerate the time-
likeRG-invarianteffectivecouplingα˜(s)andasetofpip- whereC carriesthetopmassdependenceandΠGGisthe
1
izated functions. Certainly we deal with the analytical induced vacuum polarization by the renormalized gluon
continuations [7] of Euclidean perturbative expansions operator. The gluonic decay width is factorized by the
when using the reverse transformation. This results in K-factor, Γ(H gg)=KΓ (H gg).
Born
→ →
pipizated expressions [8, 9]. The non-power sets of pip- Nowweareinapositionthatweshouldconsidergener-
izated functions and analyticized functions replace time- alizationsoftheBLMapproach. Forahistoricaloverview
like and space-like couplings which regenerate the per- of the extensions to the approachsee [14] and references
turbative series in terms of a self-consistent scheme. therein. Our emphasis here is on the single-scale exten-
sion[15] used in [3] and the multi-scale one developed in
[2]. Scale or scales are in need of n(n 1)/2 parameters
1. Higgs decay widths tobeabletoabsorballvacuumpolariz−ationinsertionsin
thenthordertruncatedseries. Thesingle-scaleextension
AsanimportantandinterestingillustrationoftheDe- uses these parameters at once. However the multi-scale
viationPatternprocedure(DPA),wefirstconsiderHiggs version takes them step by step with a specific theoreti-
bosondecays[10]intobottomquarks[11,12]andgluons calconsideration. Categorizingcorrectionsformstheba-
[13]. ThehadronicdecaywidthoftheHiggsbosonseems sis of some of the other generalizations such as [14] also.
to have large QCD corrections to the Born approxima- Taking advantage of the single-scale ℓ results in
187C3 17C2 203 C3 51C2C 121C2 r 3993C3 r 11C r
rpre = Ar Ar + Ar + A Fr A 20r A 21r A 20r
4 − 32 T2 32− 8 T 31 1152T 21 32 T 21− 8 T2 r 32− 64 T3 r 32− 2 T r 31
F F F F F 1 F 1 F 1
363C2 r C r2 55C2 r 33C C r 33C r 605C2 r2
A 21r +11 A 20r + A 20r + A F 20r A 30r + A 21r
− 16 T2 r 31 T r2 21 8 T r 21 8 T r 21− 4 T r 21 16 T2 r2 20
F 1 F 1 F 1 F 1 F 1 F 1
1331C3 r3 1089C2C r2 627C3 r2 5 3 515 155 21
+ A 21 + A F 21 + A 21 +[( C + C )r + C2r C C r C2r
32 T3 r2 64 T2 r 32 T2 r 4 A 4 F 31 576 A 21− 192 A F 21− 32 F 21
F 1 F 1 F 1
21C2 33C C 121C2 r 11C r 11C r2 173C2 r2 5 r r
+ Ar + A Fr + A 21r + A 21r A 21r A 21 C 20 21
16T 32 16 T 32 16 T2 r 32 4 T r 31− 2 T r2 20− 32 T r − 2 A r
F F F 1 F 1 F 1 F 1 1
3 r r 121C2 r3 165C C r2 r r r2 r r r
C 20 21 A 21 A F 21 [phone] +3 30 21]n
−2 F r − 16 T2 r2 − 32 T r r − r2 r f
1 F 1 F 1 1 1 1
5 3 79 11 5 3 r2 r r r2 r r r
+[( C + C )r ( C T + C T )r ( C C ) 21 [phone] +2 20 32]n2
4 A 4 F 32− 288 A F 48 F F 21− 8 A− 8 F r r − r2 r f
1 1 1 1
r3 r r
+( 21 +2 21 32)n3, (2)
− r2 r f
1 1
as the predicted 4th order coefficient. Strange thing is based prediction where the common factor r2(4T r +
1 F 20
the appearance of T in the denominator of Eq.(1) and 11C r )T3 sits as the denominator of the whole rpre.
F A 21 F 4
Eq.(2). This is even more unusual for the multi-scale Apparently in both cases rpre does not posses a nor-
4
4
mal coefficient structure. A normal structure is con- It is more evident in the β-representation of the coeffi-
structed through a combination of β-coefficients multi- cients:
plications. For re-expansion we are using a version of ℓ
rbia =81.04 259.09β +25.09β2 15.92β3+4.03β ,
which contains n(n 1)/2 parameters for the nth order 4pre − 0 0− 0 2
−
series. One suggestion might be to add n extra param- r =120.0 261.26β +75.95β2+12.96β3+4.03β .
eters to ℓ or ℓs when re-expanding to the (n+1)th or- 4 − 0 0 0 2
der series and see whether a standard structure of rpre The genuine β-representation and seBLM are developed
4
could determine these parameters. Perturbative series in [14] but here we have used a different combination of
parameters r ,r ,r presented in Eq.(1) and Eq.(2) β- coefficients for the β-representation. As we gofurther
1 2 3
{ }
intrinsically contains local symmetry group parameters andemployDPA, normalpredictionwill be closerto the
C ,C ,T ,... . So before following the above sug- calculatedresult. Sothefirstterminitsβ-representation
A F F
{ }
gestion, relations must be rewritten in terms of new se- is near r4’s. At the same time the biased prediction de-
riesparameterswhicharegroupparametersindependent. creasesitsslopeandreaches 54.5961fornf =6incom-
−
However we will get around this and take advantage of parisonto 382.9029ofthelaststepwhilethecalculated
−
the single-scale extension in the following predictions. value is 37.2079. Taking advantage of the Adler func-
−
In this way the normal BLM prediction generates the tion makes the biased prediction get away from r4 but
following series as the 4th order coefficient of R˜ compar- the normal prediction continues decreasing its slope and
ing to the calculated one: this time gets really close to r4 as nf increases:
r˜4pre=1103.4−268.480nf +10.67n2f −0.046n3f, r4pre =86.31−112.75β0+29.87β02−2.49β03+4.61β2,
r˜ =39.354 220.943n +9.685n2 0.020n3.
4 − f f − f which reads 35.2114for nf =6.
−
If we perform comparison and prediction for the corre- If we establish the ratio as(Q2)KR˜, normal prediction
sponding Adler function, we will have: willgeneratearesultdivergingfromthecalculatedcoeffi-
cientasn increases. AsweuseDPAitsuddenlychanges
r˜pre = 1627.6+191.328n 11.41n2 +0.245n3, f
4 − f − f f its orientation towards the calculated result. Utilizing
DPA with the Adler transformation will make them be-
which indicates an improvement.
have more similarly.
The normal prediction for the 4th order coefficient of
the K factor reads:
gpre=2219.878 901.698n +53.524n2 0.516n3, 2. Adler function and Bjorken sum rule
3 − f f − f
g =3372.073 866.588n +48.088n2 0.538n3.
3 − f f − f The vacuum polarization induced by vector current
Performing comparison and prediction for the corre- j = ψ¯γ ψ is the time-ordered correlationfunction
µ i i µ i
sponding Adler function improves the result:
P
gpre =3783.694 792.353n +42.653n2 0.581n3, (qµqν q2gµν)Π(Q2)=i d4xeiq.x 0 T[jµ(x)jν(0)] 0 ,
3 − f f − f − Z h | | i
whereas preventing the first term of the perturbative se- where Q2 = q2. The Adler function corresponding
ries affect the prediction during re-expansion (a biased −
to the current takes form through the scalar correlator
BLM prediction [16]) alongside the above procedure re- Π(Q2),
sults in a better behavior for higher number of flavors:
d
gpre =533.831 189.814n +15.738n2 0.318n3. D(Q2)= 12π2Q2 Π(Q2).
3 − f f − f − dQ2
Biased prediction refers to this kind of biasing.
Integration over polarized proton and neutron structure
BrieflynormalBLMpredictionforboththe decaysre- functions gp,n is related to the Bjorken polarized sum
sults in patterns whose deviations from the calculated 1
rule quantity d through
coefficients do not have serious n dependences. Spe- Bj
f
ciallyforR˜ itis soclearthatthe majorpartofdeviation 1
is due to the first part of 4th order coefficient. In the Γp1−n=Z dx(g1p(x,Q2)−g1n(x,Q2))
case of K factor we have two large parts. The deviation 0
is shared among both of them, however again the first = gA(1 d (Q2))+ ∞ µp2j−n(Q2),
part plays the main role. DPA will make r˜pre recover r˜ 6 − Bj Q2j−2
forhighervaluesofn ,whilstthishappens4forgpre when4 Xj=2
f 3
DPA is utilized by a second-class biasing. where (1 d ) denotes the coefficient function C
Bj Bjp
If we consider the ratio as(Q2)KR˜, a normal BLM pre- and gA is−the charge corresponding to the axial vector
dictionwillindicate the samebehaviorasbefore. This is current of the nucleon. Recently the coefficient func-
whilst a biased one shows a tendency between the pre- tion CBjp(Q2) and the non-singlet component of the
dicted andcalculatedcoefficients for highervalues ofn . Adler function DNS(Q2) [17] have been calculated to
f
5
order α4. They are based on the constraints due to seems that CBjp in this ratio affects parts of DNS for
s
the special form of the generalized Crewther relation which the DNS pattern was weakened while employing
Dth(easg)aCuBgejpg(arosu)p=β-1fu+ncβti(aoasns))PanidKtiahise c[1o8lo,r1s9t]ru(βct(uarse) oisf DPTAa.blIetImIIakshesowthsethneewrepsureltdsicftoironthsebreattteior fCoBrjnp(fQ2=).6.
DNS(Q2)
the coefficientsofthese functions [20]. Itisworthnotify-
ing that more recently the generalizedCrewther relation
(CR) has been expressed in terms of new β-independent TABLE III. CBjp prediction results.
polynomialsby factorizingmultiple powersofβ-function DNS
[21]: CBjp n0 n1 n2 n3 1sttermoftheβ-rep.
DNS f f f f
β(a ) n cdp4re -54.66 33.49 -2.26 0.0319 -50.64
D(as)CBjp(as)=1+ s n(as). (3) cdp4re1 -101.41 30.14 -1.67 0.0153 -66.90
nX≥1(cid:18) as (cid:19) P cdp4re2 -37.55 18.99 -1.02 0.0028 -66.90
cde4xact -172.83 45.62 -3.03 0.0379 -150.66
Adapting the old and new structures of CR reveals the
structure of coefficients.
n
PredictionPresults for CBjp and DNS are summarized Existence of DNS in this ratio absorbs the converging
in Table I. For CBjp it seems that a major part of devi- behaviorofDPApredictionforhighervalues ofn . This
f
converts it to a general improvement along all values of
n . The corresponding results for the biased BLM pre-
f
TABLE I. CBjp and DNS prediction results. diction are tabulated in Table IV. This indicates real
CBjp n0 n1 n2 n3 DNS n0 n1 n2 n3
f f f f f f f f
cp4re -265.4 95.26 5.94 0.08 dp4re 362.1 -99.08 5.04 -0.05 TABLE IV. CBjp biased prediction results
cp4re1a -261.3 67.71 -3.62 0.04 dp4re1 317.2 -93.26 4.76 -0.03 DNS
cp4re2b -20.89 26.16 -1.23 -0.01 de4xact 407.4 -103.3 5.63 -0.03 CBjp n0 n1 n2 n3 1sttermoftheβ-rep.
ce4xact -479.4 123.4 7.69 0.10 cDdNp4rSe 52.f07 13.f64 -1.f40 0.02f44 -72.32
a DPA. cdp4re1 173.65 -21.23 0.41 0 -140.29
b DPAutilizedbyAdlerfunction. cdp4re2 461.62 -73.41 3.19 -0.0409 -140.29
cde4xact -172.83 45.62 -3.03 0.0379 -150.66
ation for the normal prediction is due to the first term.
ButunlikethecaseofHiggsdecayintobottomquarksthe
improvements in the β-representation.
gapbetweenpredictionandcalculationisnotsolarge. It
iswhilstDNS iscompletelydifferent. Thepredictionand
calculatedresultsmayseemreallyclosetoeachotherbut
their β-representations are really far. There seems to be 3. Quadratic Casimir invariants
a similarity between DNS’s and Higgs decay into gluons
width’s deviationpatterns,bothofthemhaveextremum Each part in the first term of a normal BLM pattern
near nf = 3. DPA makes an improvement in CBjp pre- Eq.(2)containsatleastonefactorCA. Sofor the caseof
diction for higher values of nf and provides a slight im- HiggsdecayintobottomquarkssettingCA =0improves
provementintheβ-representation. Agoodimprovement the normalprediction significantly along all values of n
f
in the first term of β-representation for DNS is visible specifically for lower ones.
in its DPA prediction. Atlast takingadvantageofAdler Working onthe ratio a (Q2)R˜ andusing r ,r that
transformation will not improve CBjp prediction. s K { 1 2}
came out of DPA, one will find out the sensitivity of the
The results of predictions for the ratio CDBNjSp((QQ22)) could n1f part of the 4th order prediction to CA in such a way
be illustrated in Table II. The deviation pattern of the that setting CA = 0, CF = 1, and TF = 1 results in a
0.04 relative error in predicting this part.
Forthe ratioa (Q2)K aninteresting improvementoc-
TABLE II. DNS prediction results. curs when choosisng theR˜canonical i.e., conventional form
CBjp
of the quadratic Casimir invariants for SU(2). More
DNS n0 n1 n2 n3 1sttermoftheβ-rep.
CBjp f f f f freedom is obtained by departing from the conventional
dcp4re 1925.9 -463.57 24.11 -0.2872 239.47 choice to the overall normalization [22] where parame-
dcp4re1 1409.6 -332.47 16.05 -0.1302 397.27 tersbandN takecontrolofthe normalization. Thisalso
dcp4re2 111.29 -140.82 7.128 -0.0043 397.27 indicates that SU(2) is a better choice for this ratio.
dce4xact 2444.4 -572.69 31.02 -0.3415 595.05
Adapting the first part of the 4th order coefficient of
CBjp by using the overall normalization indicates that
ratiodoesnotcontainanyextremumandisdecreasingfor SU(2) improves the n2 part and SU(4) improves the n1
f f
higher values of n . DPA prediction is good for n = 6 part in comparison to SU(3) while SU(3) still provides
f f
andmakinguseofAdler transformationwillnothelp. It a better combination.
6
Averyexcitingresultisobtainedwhenperformingthe Every a observable connects to R through a specific
2
lastprocedureforDNS. InthiscaseSU(2)givesussuch channel. TemporarilyweputtheBLMpatternconstraint
a combination of n1 and n2 parts that makes the BLM aside. If we consider a to be the Adler function corre-
f f 2
pattern fit to the calculatedterms in the whole regionof spondingtoR,averysimpledependenceofthe4thorder
n . It is in spite of the slight separation occurred in all coefficient of R to both e and b is obtained. So either
f 4 4
the previous cases for higher values of n . onecanpredictboth e andb orjustpredicte anduse
f 4 4 4
Oneshouldpayattentiontotheseparametersofgauge Adler transformation for b at the same time. Perform-
4
symmetry because their effects are noticeable on the ingthe firstchoiceforR˜ resultsinapatterncrossingthe
problem and specially on an approach which seems to exactpattern aroundn =5 and for the secondchoice a
f
be related to the concept of conformal symmetry. crossing at n =2.
f
A more reasonable step would be to start with a spe-
cific type of relation between two observables and then
B. Second observable
equip the situation with a BLM pattern. Of course it
would be a waste of time if this ends to direct relations
In the previous section, we tried to find parts of the where we encounter observables whose predictions are
observable that generate a BLM pattern. It is possible equivalent. What we mean by a direct relation is a kind
to write the observable in a complete BLM pattern form oflinear relationbetweentwo variables. The situationis
in terms of an appropriate effective charge. This means clarified later. For now we should start with a Crewther
anabsolute freedomin choosingthe first two coefficients like relation between two observables R and a2:
e , e or equivalently b , b :
1 2 1 2 ∞ ∞
R=e a + e a2 + e a3 + e a4 + ... , [r0+r1as+ riasi][r0 r1as+ biasi]
1 2 2 2 3 2 4 2 −
a =b a + b a2 + b a3 + b a4 + ... . Xi=2 Xi=2
2 1 2 3 4 β K
a
= C+ , (5)
As inEqs.(1,2)we willhavethe followingrelationsforei a
coefficients:
where for the first observable we have R = r a +
1 s
e3+=(−21e1226e01eCT21FA22 +ee221154−e21C187ACT+FA2e2341e2−1CF121)nCTfFA+e2e0eee122211n2f PPamni∞i∞idn==e22Ksrbaiiana=sysii.,Pcoaβ∞ine=affid1icfsKoiertinhatteih.befias{miernc0oo.au.n.srsdpiβ}oe-ncfauieannwldcmet{ihoaKannv1n−.ee..raKP,2ci∞i−=o=n10−}sβirddi1eaearit+isen+2rg-,
1 1 all known coefficients prior to r and K ,
i i−1
C 11 33 33
A
e = [ e e + e e C e e ]
4 e1TF − 2 20 31 8 20 21 F − 4 30 21 b = r 11CAK +C j <i 1
+ CA3 [627e221 187e ] + CA3 203 e ij − ij − 12 r0 (i−1)j ij −
T2 32 e − 32 32 T 1152 21 TF
CF2 51 1 17 CF2 1089 bi(i−1)= −ri(i−1)− 3r0K(i−1)(i−2). (6)
+ A[ e C e ]+ A [ e2 C
T 32 21 F − 8 31 e T2 64 21 F Despite the fact that R does not contain r , it plays a
F 1 F 0
363 crucial role in normalizing ri coefficients in Eq.(6). To
e e ] + ..., (4)
21 31 involver in the problem,one shouldperformprediction
− 16 0
one order backward and normalize the result with the
so there is an infinite number of observables in terms of
appropriate r . Finally this will make e coefficient be
0 i
which a BLM pattern is obtained for R.
determinedintermsof r ...r . TakingK coefficients
Averyfirstchoicefore ,e wouldber ,r . ForR˜ the { 1 i−1}
1 2 1 2 as the free variables,Eq.(6) reduces into the very simple
predicted result suffers the same problem as the normal
direct relation b = r +f (n ). The result comes out
i i i f
BLM prediction i.e., a large deviation due to the first −
of such a relation suffers the same problem as a normal
term. An impossible state would be the generation of
prediction for r since b and r can exchange roles. The
i i i
both e and b through BLM pattern.
same is true about predicting r through e . Although
i i
Obviously we cannot use strict constraints to obtain
we are able to determine e prior to r using Eq.(5), at
i i
some b , b because these cannot be partners to the first
1 2 the same time e has a direct relation with r i.e., e =
i i i
constraint due to the BLM pattern. For instance if a
2 c r +h (n ). So if anything is to give us real different
i i i f
is determined as the Adler function corresponding to R,
results, it must have an indirect relation with r :
i
e will not be able to normalize R appropriately. It is
1
enslaved to take r1 while it must play the role of an im- i−1
portant building block in the BLM pattern at the same bij= [lijrij +mijK(i−1)(j−1)]+Cij,
time. Generating all b coefficients through Adler func- Xj=0
tionexceptb1 andb2 wouldbe a muchmoreflexible way i−1
which reduces the infinite set of a2 observables to a set ei = cijrij +hi(nf).
of few a2s. Xj=0
7
We have done a kind of converting such direct relations is clear that involving ith order K coefficients in Eq.(6)
into indirect ones in the past when we tried to interpret insteadof (i 1)th orderones couldbe obtained by con-
−
DPAaschangingweightsofrij coefficientsinthenormal sidering βaa2K in Eq.(5):
BLM pattern. Obviously Eq.(6) is also a kind of indi-
rectrelationbetween r and K . Following this line, it
i i−1
∞ ∞
β K
[r +r a+(r +r β )a2+ r ai][r r a+(b +b β )a2+ b ai] = C+ a , (7)
0 1 20 21 0 i 0− 1 20 21 0 i a2
Xi=3 Xi=3
where we’ve written all the coefficients in the β- III. CONCLUSIONS
representationso r =r β + i−1r βj. The
i (i−1)(i−2) i−2 j=0 ij 0
intrinsicpropertyofthisconstraintwoulPdbetoeliminate Flexibility of any optimization prescription in pertur-
K0,K1,K21,K32,K34 from the pattern. It is possible bative analysis is the key to check the possibility of im-
{ }
to use Eq.(7) alongside Eq.(5). When βaK get involved proving the prediction by taking deviation of lower or-
a
in Eq.(7), an oversimple indirect relation appears. The der predictions into account i.e., finding patterns for the
indirectness comes from the absence of K parts in b : perturbative series. Taking DPA as finding parts of the
i i0
observable that generates the appropriate BLM pattern
b = r +D
i0 i0 ij for each order might help us in categorizing observables
−
Ki(j−1) andtheirresponsestothe predictionrespectingtheirde-
b = r +E 0<j <i 1
ij − ij − r ij − viation patterns. But it could not be fully utilized to
0
K improve the prediction since for any order n we will suf-
i(i−2)
bi(i−1)= −ri(i−1)− r0 . (8) ftehreleaffckecotfoifntfhoermsyamtiomnetarbyogurto(unp−pa1r)apmaertaemrseotenrst.heInBfLaMct
As in Eq.(6) we combined all lower order coefficients’ pattern is much more noticeable where the overall nor-
dependences in constant parts D and E . Again pre- malizationplaysakeyrole. Areasonablewaytoimprove
ij ij
diction will depend on r which should be resolved like predictedpatterns is to performpredictionsbasedonin-
0
before. direct relations. Observables are connected in a direct
One could rewrite Eq.(6) in terms of the coefficients channel when they are expressed in terms of each other.
ofthe β-independentpolynomials inEq.(3)usingthe In this case even specific insightful constraints such as
n
P
relations in [21]. The importance of doing this is that the Crewther relation does not guarantee a simple indi-
the idea in [21] strongly reveals existence of flavor-inde- rect connection between observables. As a note for the
pendent polynomials . Conformalsymmetry breaking reader,clearlywearenotreferringtothephysicalcontent
n
P
totally happens through β-function in this way. So it ofCrewtherrelationbutjustthemathematicalformofit
is more appropriate to rewrite Eq.(6) in terms of these as a connection channel. So finding a mother constraint
polynomial coefficients, even though this does not affect thatproduceselegantindirectrelationsisthechallenging
the directness of Eq.(6). problem in the context of finding patterns.
[1] S.J.Brodsky,G.P.Lepage, andP.B.Mackenzie,Phys. [9] D. V. Shirkov, Theor. Math. Phys. 127, 409 (2001),
Rev.D28, 228 (1983). arXiv:hep-ph/0012283.
[2] S.J.BrodskyandH.J.Lu,Phys.Rev.D51,3652(1995), [10] A. Djouadi, Phys. Rept. 457, 1 (2008),
arXiv:hep-ph/9405218. arXiv:hep-ph/0503172.
[3] S. J. Brodsky, G. T. Gabadadze, A. L. Kataev, [11] P. A. Baikov, K. G. Chetyrkin, and J. H. Kuhn, Phys.
and H. J. Lu, Phys. Lett. B372, 133 (1996), Rev. Lett.96, 012003 (2006), arXiv:hep-ph/0511063.
arXiv:hep-ph/9512367. [12] A. L. Kataev and V. T. Kim, (2008),
[4] T.vanRitbergen,J.A.M.Vermaseren, andS.A.Larin, arXiv:0804.3992 [hep-ph].
Phys.Lett. B400, 379 (1997), arXiv:hep-ph/9701390. [13] P. A. Baikov and K. G. Chetyrkin,Phys. Rev. Lett. 97,
[5] P.Cvitanovic, Phys.Rev. D14, 1536 (1976). 061803 (2006), arXiv:hep-ph/0604194.
[6] M. Mojaza, C. Pica, and F. Sannino, (2010), [14] S. V. Mikhailov, JHEP 06, 009 (2007),
arXiv:1010.4798 [hep-ph]. arXiv:hep-ph/0411397.
[7] D. V. Shirkov and I. L. Solovtsov, (1996), [15] G. Grunberg and A. L. Kataev, Phys. Lett. B279, 352
arXiv:hep-ph/9604363. (1992).
[8] D.V. Shirkov, (2000), arXiv:hep-ph/0003242.
8
[16] A. Mirjalili, B. A. Kniehl, and M. R. Khellat, (2010), [20] P. A. Baikov, K. G. Chetyrkin, and J. H. Kuhn, Phys.
(privatecommunication). Rev.Lett.104,132004(2010),arXiv:1001.3606 [hep-ph].
[17] P.A.Baikov,K.G.Chetyrkin, andJ.H.Kuhn, (2010), [21] A. L. Kataev and S. V. Mikhailov, (2010),
arXiv:1007.0478 [hep-ph]. arXiv:1011.5248 [hep-ph].
[18] D. J. Broadhurst and A. L. Kataev, Phys. Lett. B315, [22] T. van Ritbergen, A. N. Schellekens, and J. A. M.
179 (1993), arXiv:hep-ph/9308274. Vermaseren, Int. J. Mod. Phys. A14, 41 (1999),
[19] R. J. Crewther, Phys. Lett. B397, 137 (1997), arXiv:hep-ph/9802376.
arXiv:hep-ph/9701321.