Exact Amplitude-Based Resummation in Quantum Field Theory: Recent Results

0.13 MB·English

by  B. F. L. Ward

Exact Amplitude-Based Resummation in Quantum
Field Theory: Recent Results
2
1
0
2

n
a
J B.F.L.Ward∗†

2 BaylorUniversity
E-mail:[email protected]
]
h
S.K. Majhi
p
- IndianAssociationfortheCultivationofScience
p
e E-mail: [email protected]
h
[ S.A. Yost‡

TheCitadel
1
v E-mail: [email protected]
5
1
5 Wepresentthecurrentstatusoftheapplicationofourapproachofexactamplitude-basedresum-
0 mationin quantumfield theoryto two areasofinvestigation: precisionQCD calculationsof all
.
1 threeofusasneededforLHCphysicsandtheresummedquantumgravityrealizationbyoneofus
0
2 (B.F.L.W.)ofFeynman’sformulationofEinstein’stheoryofgeneralrelativity.Wediscussrecent
1 resultsastheyrelatetoexperimentalobservations. Thereisreasonforoptimismintheattendant
:
v comparisonoftheoryandexperiment.
i
X
r
a
10thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumFieldTheoryto
Phenomenology)
September26-30,2011
Mamallapuram,India
BU-HEPP-11-04,
Dec.,2011
∗Speaker.
†WorksupportedinpartbyD.o.E.grantDE-FG02-09ER41600.
‡WorksupportedinpartbyD.o.E.grantDE-PS02-09ER09-01andgrantsfromTheCitadelFoundation.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ExactAmplitude-BasedResummationinQuantumFieldTheory: RecentResults B.F.L.Ward
1. Introduction
Withthestart-upoftheLHCtheeraofprecisionQCD,bywhichwemeanpredictionsforQCD
processes at the total precision tag of 1% or better, is upon us and the need for exact, amplitude-
based resummation of large higher order effects is paramount. Such resummation allows one to
have better than 1% precision asarealistic goal asweshall show in whatfollows, sothat one can
indeed distinguish new physics(NP) from higher order SM processes and can distinguish differ-
ent models of new physics from one another as well. In a parallel development, the issue of the
application of ordinary quantum field theoretic methods to Einstein’s theory of general relativity
lends itself as well to a resummation approach, provided again that the resummation is an exact
amplitude-based one, asone ofus(B.F.L.W.)has shown. Inwhatfollows, wepresent the status of
thesetwoapplications ofexactamplitude-based resummationtheoryinquantum fieldtheory.
Thetwoparadigmswhichwepresentarethenasfollows. First,inthenextSection,wepresent
an approach toprecision LHCphysics which isan amplitude-based QED⊗QCD(≡QCD⊗QED)
exact resummation theory [1] realized by MCmethods. Thestarting point isthen the well-known
fullydifferential representation
ds =(cid:229) dx dx F(x )F (x )dsˆ (x x s) (1.1)
1 2 i 1 j 2 res 1 2
Z
i,j
of a hard LHC scattering process using a standard notation so that the {F } and dsˆ are the
j res
respective parton densities and reduced hard differential cross section where we indicate the that
latterhasbeenresummedforalllargeEWandQCDhigherordercorrectionsinamannerconsistent
withachieving atotal precision tag of1% orbetter forthe total theoretical precision of(1.1). The
key issue to precision QCD theory is then the determination of the value of the total theoretical
precision of(1.1),whichwedenotebyD s . Itcanbedecomposed asfollows:
th
D s =D F⊕D sˆ (1.2)
th res
inanobviousnotationwhereD AisthecontributionoftheuncertaintyonAtoD s . Thetheoretical
th
precision D s validates theapplication ofagiven theoretical prediction toprecision experimental
th
observations, for the discussion of the signals and the backgrounds for both SM and NP studies,
andmorespecificallyfortheoverallnormalization ofthecrosssectionsinsuchstudies. NPcanbe
missed if acalculation with an unknown value of D s isused for such studies. This point cannot
th
beemphasized toomuch.
By our definition, D s is the total theoretical uncertainty coming from the physical preci-
th
sion contribution and the technical precision contribution [2]: the physical precision contribution,
D s phys, arises from such sources as missing graphs, approximations to graphs, truncations,....;
th
the technical precision contribution, D s tech, arises from such sources as bugs in codes, numerical
th
rounding errors, convergence issues, etc. Thetotaltheoretical erroristhengivenby
D s =D s phys⊕D s tech. (1.3)
th th th
Thedesired value forD s depends onthespecific requirements ofthe observations. Asageneral
th
rule, one would like that D s ≤ fD s , where D s is the respective experimental error and
th expt expt
2
ExactAmplitude-BasedResummationinQuantumFieldTheory: RecentResults B.F.L.Ward
f . 1 so that the theoretical uncertainty does not significantly affect the analysis of the data for
2
physicsstudies inanadverseway.
With the goal of achieving such precision in a provable way, we have developed the QCD⊗
QEDresummationtheoryinRefs.[1]forthereducedcrosssectionin(1.1)andfortheresummation
oftheevolutionofthepartondensitiesthereinaswell. Inbothcases,thestartingpointisthemaster
formula
(cid:213) mj2d=s1¯rdek3s′kb=j′˜¯2j2eRS(Uk(2dMp4,Iy)R.4.(eQ.i,Cyk·E(Dp;1)k+(cid:229)′q,1¥n.−,.m.p=2,−0k′qn2!)1m−d!(cid:229)3Rpk2j(cid:213)1d−3nj(cid:229)q1=2k,′1j2d)k3+kj1Dj1QCED (1.4)
n,m 1 n 1 m p0 q0
2 2
whereds¯ iseitherthereducedcrosssectiondsˆ orthedifferentialrateassociatedtoaDGLAP-
res res
CS [3,4] kernel involved in the evolution of the {F } and where the new (YFS-style [5]) non-
j
Abelian residuals b˜¯ (k ,...,k ;k′,...,k′ ) have n hard gluons and m hard photons and we show
n,m 1 n 1 m
thefinalstatewithtwohardfinalpartonswithmomenta p , q specifiedforageneric2f finalstate
2 2
fordefiniteness. TheinfraredfunctionsSUM (QCED), D aredefinedinRefs.[1,6,7]. This
IR QCED
simultaneousresummationofQEDandQCDlargeIReffectsisexact. Moreover,theresidualsb˜¯
n,m
allowarigorous partonshower/MEmatchingviatheirshower-subtracted counterparts bˆ˜¯ [1].
n,m
Theresultin(1.4)alsoallowsusananexact,amplitude-basedresummationapproachtoFeyn-
man’s formulation of Einstein’s theory, as one of us (B.F.L.W.) has shown in Refs. [8] via the
followingrepresentation oftheFeynmanpropagators inthattheory:
i
iD ′ (k)=
F (k2−m2−S +ie )
s
ieB′g′(k)
=
(k2−m2−S ′+ie )
s
≡iD ′ (k)| .
F resummed
for scalar fields with an attendant generalization for spinning fields [8]. We stress that there are
no approximations in (1.5). The formula for B′′(k) is given in Refs. [8] and is presented below.
g
Wenow discuss the two paradigms opened by (1.4) for precision QCDfor the LHCand for exact
resummation ofEinstein’s theoryinturn.
2. PrecisionQCDforthe LHC
We first stress that the methods we emply for resummation of the QCD theory are fully con-
sistent with the methods in Refs. [9,10]. This can be seen by considering the application of the
latter methods to the 2→ n processes [f] at hard scale Q, f (p ,r )+ f (p ,r )→ f (p ,r )+
1 1 1 2 2 2 3 3 3
f (p ,r )+···+ f (p ,r ),wherethe p,r label4-momentaandcolorindicesrespectively,
4 4 4 n+2 n+2 n+2 i i
byAbyatetal. inRef.[11],wheretherespectiveamplitude isrepresented as
C
M[f] =(cid:229) M[f](c )
{ri} L L {ri}
L
(2.1)
C
=J[f](cid:229) S H[f](c ) ,
LI I L {ri}
L
3
ExactAmplitude-BasedResummationinQuantumFieldTheory: RecentResults B.F.L.Ward
where repeated indices are summed, and the functions J[f],S , and H[f] are respectively the jet
LI I
function, thesoftfunction whichdescribes theexchange ofsoftgluons betweentheexternal lines,
andthehardcoefficientfunction. Thelatterfunctions’infraredandcollinearpoleshavebeencalcu-
lated to 2-loop order in Refs. [11]. Tomake contact between eqs.(1.4,2.1), identify inthe specific
application Q¯′Q → Q¯′′′Q′′+m(G) in (1.4) f = Q,f = Q¯′,f = Q′′,f = Q¯′′′,{f ,···,f } =
1 2 3 4 5 n+2
{G ,···,G }, in (2.1), where we use the obvious notation for the gluons here. This means that
1 m
n=m+2. Then,touseeq.(2.1)ineq.(1.4), oneobservesthefollowing:
I. Byits definition ineq.(2.23) ofRef. [11], the anomalous dimension of the matrix S does not
LI
contain any of the diagonal effects described by our infrared functions SUM (QCD) and
IR
D ,where
QCD
SUM (QCD)=2a ´ B +2a B˜ (K ),
IR s QCD s QCD max
d3k
2a B˜ (K )= S˜ (k)q (K −k),
s QCD max Z k0 QCD max
d3k
D = S˜ (k) e−iy·k−q (K −k) , (2.2)
QCD QCD max
Z k
h i
wherethereal IRemission function S˜ (k)and thevirtual IRfunction ´ B aredefined
QCD QCD
eqs.(77,73) inRef.[6]. Notethat(1.4)isindependent ofK .
max
II.Byitsdefinitionineqs.(2.5)and(2.7)ofRef.[11],thejetfunctionJ[f]containstheexponential
ofthevirtualinfrared function a ´ B ,sothatwehavetotakecarethatwedonotdouble
s QCD
countwhenweuse(2.1)in(1.4)andintheequations inRefs.[1,6,7]thatleadthereto.
In this way we get the following realization of our approach using the results in Ref. [11]: In our
result in eq.(75) in Ref. [6] for the contribution to (1.4) of m-hard gluons for the process under
studyhere,
dsˆm= e2a s´ BQCD (cid:213) m d3kj d (p +q −p −q −(cid:229)m k)
m! Z (k2+l 2)1/2 1 1 2 2 i
j=1 j i=1
d3p d3q
r¯(m)(p ,q ,p ,q ,k ,···,k ) 2 2, (2.3)
1 1 2 2 1 m p0q0
2 2
wecanidentify theresidual r¯(m) asfollows:
r¯(m)(p ,q ,p ,q ,k ,···,k )=(cid:229) |M[f]|2
1 1 2 2 1 m colors,spin {ri}
≡spins,(cid:229){ri},{ri′}hc{sri}{ri′}|J¯[f]|2L(cid:229)C=1L(cid:229)′C=1SL[fI]HI[f](cL){ri}(cid:16)SL[f′I]′HI[′f](cL′){ri′}(cid:17)†, (2.4)
where here we defined J¯[f] =e−a s´ BQCDJ[f], and we introduced the color-spin density matrix for
the initial state, hcs, so that hcs = hcs , suppressing the spin indices, i.e., hcs only
{ri}{ri′} {r1,r2}{r1′,r2′}
depends on the initial state colors and has the obvious normalization implied by (2.3). Proceed-
ing then according to the steps in Ref. [6] leading from (2.3) to (1.4) restricted to QCD, we get
the corresponding implementation of the results in Ref. [11] in our approach, without any double
countingofeffects. Thisprovesthatthenewnon-Abelianresidualsb˜¯ in(1.4)transcendthoseof
m,n
anAbelianmasslessgaugetheoryasintroduced inRef.[5].
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ExactAmplitude-BasedResummationinQuantumFieldTheory: RecentResults B.F.L.Ward
AswehaveexplainedinRefs.[1],thesenewnon-Abelianresidualsallowrigorousshower/ME
matching viatheirshowersubtracted analogs:
b˜¯ →bˆ˜¯ (2.5)
m,n m,n
wherethebˆ˜¯ havehadalleffectsintheshowersassociated tothe{F }removedfromthem.
m,n j
Whentheformulain(1.4)isapplied tothecalculation ofthekernels, P ,intheDGLAP-CS
AB
theory itself, wegetanimprovement oftheIRlimitofthesekernels, anIR-improved DGLAP-CS
theory [6,7] in which large IR effects are resummed for the kernels themselves. The resulting
newresummedkernels, Pexp asgiveninRef.[6,7]andasillustrated below,yieldanewresummed
AB
schemeforthePDF’sandthereduced crosssection:
F , sˆ →F′, sˆ′for
j j
1+(1−z)2
Pgq(z)→Pgeqxp(z)=CFFYFS(gq)e12dq z zgq,etc.,
withthesamevaluefors in(1.1)withimprovedMCstability asdiscussed inRef.[12]. Here,the
YFS [5] infrared factor is given by FYFS(a)=e−CEa/G (1+a) where CE is Euler’s constant and
we refer the reader to Ref. [6,7] for the definition of the infrared exponents g , d as well as for
q q
exp
thecomplete setofequations forthenewP . C isthequadratic Casimirinvariant forthequark
AB F
colorrepresentation.
The basic physical idea underlying the new kernels is illustrated in Fig. 2 as it was already
shown by Bloch and Nordsieck [13]: an accelerated charge generates a coherent state of very
G(z−Pjξj)
☛(cid:0)
✡✁
☛(cid:0)
✡✁
☛(cid:0)
✡✁
☛(cid:0)
✡✁
☛(cid:0)
SoftGluonCloud ☛✡(cid:0)✁
✡✁
zG1(ξ1)}|Gℓ(ξℓ{) ☛☛✡(cid:0)(cid:0)✁
✡✁
☛(cid:0) ··· ☛(cid:0) ☛(cid:0)
✡✁ ✡✁ ✡✁
☛(cid:0) ☛(cid:0) ☛(cid:0)
✲ ✡✈✁ ✡✈✁ ✡✈✁ ✲
q q(1−z)
q→q(1−z)+G⊗G1···⊗Gℓ, ℓ=0,··· ,∞
Figure1: Bloch-Nordsiecksoftquantaforanacceleratedcharge.
soft massless quanta oftherespective gaugefieldsothatonecannot know whichoftheinfinityof
possiblestatesonehasmadeinthesplittingprocessq(1)→q(1−z)+G⊗G ···⊗G , ℓ=0,···,¥
1 ℓ
illustrated inFig.2. Thenewkernels takethiseffectintoaccount.
The new MC Herwiri1.031 [12] gives the first realization of the new IR-improved kernels in
the Herwig6.5 [14] environment. Here, we compare it with Herwig6.510, both with and without
the MC@NLO [15] exact O(a ) correction, in Fig. 2 in relation to D0 data [16] on the Z boson
s
p in single Z production and the CDF data [17] on the Z boson rapidity in the same process all
T
at the Tevatron. We see [12] that the IR improvement improves the c 2/d.o.f in comparison with
the data in both cases for the soft p data and that for the rapidity data it improves the c 2/d.o.f
T
5
ExactAmplitude-BasedResummationinQuantumFieldTheory: RecentResults B.F.L.Ward
(a) (b)
Vector boson rapidity Generated Z transverse momentum.
0.6 66 < MZ/g* < 116, plt > 20 0.12 DGLAP
0.11
DGLAP−CS
0.5 IR.Imp.DGLAP−CS −1c) 0.1 IR.Imp.DGLAP
V/0.09
0.4 e
G
0.3 (dpt00..0078
/
sd0.06
0.2
· 0.05
s1/
0.1 0.04
0.03
00 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14
Y(Z) Z/g* p (GeV/c)
t
Figure 2: From Ref. [12], comparison with FNAL data: (a), CDF rapidity data on (Z/g ∗) production to
e+e− pairs, the circular dots are the data, the light(dark) lines are HERWIG6.510(HERWIRI1.031); (b),
D0 p spectrum data on (Z/g ∗) production to e+e− pairs, the circular dots are the data, the dark trian-
T
gles are HERWIRI1.031, the light triangles are HERWIG6.510. In both (a) and (b) the dark squares are
MC@NLO/HERWIRI1.031,andthelightsquaresareMC@NLO/HERWIG6.510,whereMC@NLO/Xde-
notes the realization by MC@NLO of the exact O(a ) correction for the generatorX. These are untuned
s
theoreticalresults.
before the application of the MC@NLO exact O(a ) correction and that with the latter correc-
s
tion the c 2/d.o.f’s are statistically indistinguishable. Moreimportantly, this theoretical paradigm
can be systematically improved in principle to reach any desired D s . The suggested accuracy
th
at the 10% level shows the need for the NNLO extension of MC@NLO, in view of our goals
for this process. We are currently developing the analogous applications for the new kernels for
Herwig++ [18], Herwiri++, for Pyhtia8 [19] and for Sherpa [20]. In addition we are currently
analysizing recent LHCdata using Herwiri1.031/MC@NLO, Herwiri++/Powheg [21] as weshall
reportelsewhere[22].
3. Resummed Quantum Gravity
Oneofus(B.F.L.W.)hasrecentlycontinuedhisapplicationofexactamplitude-basedresumma-
tion theory to Feynman’s formulation of Einstein’s theory, as described in Refs. [8]. In particular,
inRef.[23],hehasarrivedatafirstprinciples prediction ofthecosmological constant thatisclose
totheobservedvalue[24,25],r L ∼=(2.368×10−3eV(1±0.023))4,aswenowrecapitulate.
InRef.[23],usingthedeepUVresult
k 2|k2| m2
B′′(k)= ln , (3.1)
g 8p 2 (cid:18)m2+|k2|(cid:19)
itisshownthattheUVlimitofNewton’sconstant, G (k),isgivenby
N
360p
g = lim k2G (k2)= ∼=0.0442, (3.2)
∗ N
k2→¥ c2,eff
6
ExactAmplitude-BasedResummationinQuantumFieldTheory: RecentResults B.F.L.Ward
where[8,23]c ∼=2.56×104 fortheknownworld. Inaddition,itisshownthatthecontribution
2,eff
ofascalarfieldtoL is
d4k (2k2)e−lc(k2/(2m2))ln(k2/m2+1)
L =−8p G 0
s N2R(2p )4 k2+m2
1 (3.3)
∼=−8p G ,
N(cid:20)G264r 2(cid:21)
N
where r =ln 2 and we have used the calculus of Refs. [8,23]. We note that the standard equal-
l
c
time (anti-)commutation relations algebra realizations then show that a Dirac fermion contributes
−4timesL toL . ThedeepUVlimitofL thenbecomes
s
L (k)−→k2l ,
∗
k2→¥
l ∗=−c228,e8f0f (cid:229) (−1)Fjnj/r 2j (3.4)
j
∼=0.0817
whereF isthefermionnumberof jandr =r (l (m )). Ourresultsfor(g ,l )agreequalitatively
j j c j ∗ ∗
withthoseinRefs.[26,27].
For reference, we note that, if we restrict our resummed quantum gravity calculations above
forg ,l tothepuregravitytheorywithnoSMmatterfields,wegettheresults
∗ ∗
g =.0533, l =−.000189.
∗ ∗
We see that our results suggest that there is still significant cut-off effects in the results used for
g , l 1 in Refs. [26,27], which already seem to include an effective matter contribution when
∗ ∗
viewed from our resummed quantum gravity perspective, as an artifact of the obvious gauge and
cut-off dependencies of the results. Indeed, from a purely quantum field theoretic point of view,
thecut-offactionis
1
D S(h,C,C¯;g¯)= <h,Rgravh>+<C¯,RghC> (3.5)
k 2 k k
where g¯ is the general background metric, which is the Minkowski space metric h here, andC,C¯
aretheghostfieldsandtheoperators Rgrav, Rgh implementthecoursegraining astheysatisfythe
k k
limits
lim R =0,
k
p2/k2→¥
lim R →Z k2,
k k
p2/k2→0
for some Z [26]. Here, the inner product is that defined in the second paper in Refs. [26] in its
k
Eqs.(2.14,2.15,2.19). The result is that the modes with p.k have a shift of their vacuum energy
bythecut-off operator. Thereisnodisagreement inprinciple between ourgaugeinvariant, cut-off
independent resultsandthegaugedependent, cut-offdependent resultsinRefs.[26,27].
1InthefirstpaperinRef.[27],(g∗,l ∗)≈(0.27,0.36).
7
ExactAmplitude-BasedResummationinQuantumFieldTheory: RecentResults B.F.L.Ward
3.1 AnEstimateofL
ToestimatethevalueofL today,wetakethenormal-ordered formofEinstein’s equation,
:Gmn :+L :gmn :=−8p GN :Tmn :. (3.6)
Thecoherent state representation ofthethermal density matrixthen givestheEinstein equation in
the form of thermally averaged quantities with L given by our result above in lowest order. Tak-
ing the transition time between the Planck regime and the classical Friedmann-Robertson-Walker
regimeatt ∼25t fromRefs.[27],weintroduce
tr Pl
L (t )
r L (ttr)≡ 8p G t(rt )
N tr
(3.7)
= −MP4l(ktr)(cid:229) (−1)Fnj
64 r 2
j j
andusethearguments inRefs.[28](t isthetimeofmatter-radiation equality) toget
eq
r L (t0)∼= −MP4l(1+c2,eff6k4t2r/(360p MP2l))2(cid:229) (−1r)2Fnj
j j
t2 t2/3
× tr ×( eq )3
(cid:2)te2q t02/3 (cid:3) (3.8)
−M2 (1.0362)2(−9.197×10−3)(25)2
∼= Pl
64 t2
0
∼=(2.400×10−3eV)4.
wherewetaketheageoftheuniversetobet ∼=13.7×109 yrs. Inthelatterestimate,thefirstfactor
0
in the square bracket comes from the period from t to t (radiation dominated) and the second
tr eq
factorcomesfromtheperiodfromt tot (matterdominated)2. Thisestimateshouldbecompared
eq 0
withtheexperimental result[24,25]3 r L (t0)|expt ∼=(2.368×10−3eV(1±0.023))4. Inclosing, two
ofus(B.F.L.W.,S.A.Y.)thankProf. IgnatiosAntoniadisforthesupportandkindhospitality ofthe
CERNTHUnitwhilepartofthisworkwascompleted.
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ExactAmplitude-BasedResummationinQuantumFieldTheory: RecentResults B.F.L.Ward
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9