Magic Spinor Product Methods in Loop Integralshep.baylor.edu/hatake/HEPseminar/talks/Ward_20111121_Mag...11/21/2011 33 More comparison -- New Trace Over Fermion Lines Method (Glover&Tejeda-Yeomans)

11/21/201111/21/2011 1

Magic Spinor Product Methods

in Loop Integrals

BFL Ward

Baylor University

11/21/2011

11/21/201111/21/2011 2

Preface

Why Do We Need Such Methods?

Crucial Step Toward Resolving

Fundamental Outstanding Issues:

- Multi-Leg/Multi-Loop Quantum

Corrections with Practical

Interface to Efficient MC Evt Gen

*Amplitude-Based

FOR PRECISION LHC PHYSICS

11/21/201111/21/2011 3

State of the Art(SOTA)

- One-Loop, Multi-Leg(n>4): OPP, UC,

DUC, GOLEM, DD, Fleischer et al.,…

- Two-Loop, 2-Legs: Tarrach, Fleischer et al., .….

- Two-Loop, n-Legs, n>2: Berends et al., Bern, Dixon, Kosower,

Catani et al, .…

- n-Loop, 2-Legs, n>2: Tarasov, Larin, Chetyrkin et al., Harlander et

al., …..

- Multi-Loop, Multi-Leg: Gehrmann et al., Baikov et al., ….

Much Theory Effort: New Physics as Bounty when SM Calculated

Precisely

>

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*An Approach to Precision LHC

Physics Theory:

QEDQCD Resummation Realized

by MC Methods --

ˆ)()(,

2121

ji

resji dxFxFdxdxd

11/21/201111/21/2011 5

(1) ),,;,,(~

*

)2(!!

1

ˆˆ

0

2

2

3

0

2

2

3''

11,

)(

4

4

1

3

1

3

0,

)( 1 2

'

212211

2 2

2

1 1

1

q

qd

p

pdkkkk

eyd

k

kd

k

kd

nme

dd

nmnm

Dkkqpqpiyn

j j

jm

j j

j

nm

QCEDSUM

n

nres

j

QCED

j

jj

IR

11/21/201111/21/2011 6

Shower/ME Matching:

IR-Improved DGLAP-CS Theory:

New resummed scheme for PAB, reduced cross

section --

residuals subtracted-shower ,~̂~

,, nmnm

stability. MC improved with, for value same the giving

etc., CPP

for F F

F

exp

qqqq

jj

,)1(1

1)(

ˆ,ˆ,

2

2

1

qq

zz

zeF qYFS

11/21/2011 7

HERWIRI1.031: First Realization in HERWIG6.5

Environment--PRD81 (2010) 076008:

2

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Environment--PRD81 (2010) 076008:

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Compare

CMS(Mans)

(soon)


Environment--PRD81 (2010) 076008 – HERWIG++(SOON)

11/21/2011 10

Compare

CMS(1110.4973)

PRELIMINARY

(A.M.)


Environment--PRD81 (2010) 076008 – HERWIG++(SOON)

11/21/2011 11

Compare

ATLAS(1107.2381)

(soon)

HERWIRI1.031: PRD81 (2010) 076008 –

HERWIG++(SOON, thanks to Mike and Bryan)

11/21/2011 12

Goals in Today’s Talk

Recent Progress in Realizing Efficiently:

BFLW: CERN-PH-TH/2011-029, Phys. Rev. D

83 (2011) 113014.

1. Multi-leg Loop Corrections Essential for

O(s2, s,

2) Exactness in

2. How to Realize in (1) via MC Methods on

Event-by-Event Basis? Compactness,

Numerical Stability, Rigorousness

~,n m

~,n m

11/21/2011 13

Multi-leg Loops

res

res

Prototypical

example

11/21/2011 14

Multi-leg LoopsIssues :

OPP, UC, DUC, ….

Numerical Stability –

with

I = det(Yi)/det(Y), Yab = ma2+mb

2-(pa- pb)2

Automation:

OPP(Frixione et al., JHEP1105(2011)044) –

speed/resources

)(4

0

00 iDEi

i

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Possible Improvements:

HG Representation of All Orders -Expansion

Yost et al., PoS ICHEP2010:135,2010, refs.

therein

Chinese Magic in Loop Integrals:

11/21/2011 16

Magic: Zu et al., KS,…: for u() | >,

11/21/2011 17

Magic: HISTORY

In Z. Xu, D.-H. Zhang, and L. Chang, Nucl.

Phys. B291 (1987) 392; Tsingua University

preprint TUTP-84/3, 1984, GAUGE

INVARIANCE USED TO SHOW

ELIMINATION OF REAL RADIATION FROM

ENTIRE LINE WITH THIS REPRESENTATION

COMPACT FORMULA MAGIC!!

11/21/2011 18

Magic: note explicitly as well

( , ) ( ) / ,

( , ) ( ) / ( ),

( , )csc '

'

csc

'cot ,

( , ) csc csc csc csc

cotcsc '

'

csc

'(cot cot sec )

Q p p s

z z p p s

x xp p p

s

p p

s

y y kp p

s

p p p

s

p p

1 0

0

0

0

1 2

1 2

1 1 2 1 2

1 2

1 1 2 1

2

0 11

1

0

s

kk

kp

p

p

,

, ''

'

11/21/2011 19

Required Integral for Fig. 1c:

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Real Correction Magic: Some diagrams

vanish for appropriate choice of and

what remains is compact expression in

terms of spinor products:

<p-|q+> = <q+|p->*=

Can we do the same for virtual

corrections?

p q e q p ei iq p

11/21/2011 21

Key observation: q=ii where, unlike

in OPP, van Hameran et al., …, we use

explicit realization of the i in terms of

the external momenta {pi , k}:

11/21/2011 22

so that, explicitly,

11/21/2011 23

Bridge to loop magic:

, ~ , ~

33

44

2 2

Each part of the loop momentum now

imposes helicity conservation :

For the i = pi part, we move from pi

to pi in spinor space, i=1,2

For the 3,4 parts, we move instead

from p1 to p2 and from p2 to p1 in

spinor space

-- this is the difference with massless

4-vectors.

11/21/2011 24

11/21/2011 25

For N in eq.(1) we get now simply

where

Direct Reduction: Putting N in (1)

representation with the scalar

integrals

One scalar 5-pt fn and known 4-pt,

3-pt and 2-pt fns, as desired, no

Gram’s, etc.

11/21/2011 26

But, still one scalar 5-pt fn!

Solution:

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‘’Gram’’ Free E0 :

11/21/2011 28

where

and

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Magic Effects: Killings

All but one set of terms with 3 j .

In numerator of propagator

(before)after real emission vertex,

terms associated with and half

of terms in virtual momentum

expansion in the former case.

11/21/2011 30

( ) p k1

Magic Effects: Killings

Note as well(more magic):

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How does it compare?

Traditional Trace over Fermion Lines:

Compare 2Re MB*M(1c) where in Chinese magic we have

here with the trace of the product of the usual representation of these amplitudes which involves 2 times the product of the trace of 10 Dirac gamma matrices and the trace of 6 Dirac gamma matrices, or

2 9 7 5 4 5 4 = 50,400 terms,

which have still to undergo Passarino-Veltman reduction of 3-,2-,1-0-tensor scalar reduction.

11/21/2011 33

More comparison -- New Trace Over Fermion Lines Method (Glover&Tejeda-

Yeomans)

* Represent the amplitude M in terms of gauge invariant tensor structures {Tj} with scalar form factor coefficients {Aj} and project out the scalar from factors by projection operators P(Aj), which will then generate a trace of a linear combination of the conjugates of the Tj in product with M.

But, one of the gauge invariant tensor structures is the Born tensor structure so that our estimate above is a lower limit

on the number of terms that would obtain in this case.

Ex: for the , the projection on to the M+-++ amplitude scalar coefficient for the box graph generates a trace of 12 Dirac gamma matrices, repeated 5 times for 51,975 terms

qq

11/21/2011 34

Relationship to Other Approaches

del Aguila-Pittau(JHEP0407(2004)017):

light-like basis for loop 4-momentum

to get recursion relation between

one-loop n-point tensor integrals of

differing rank --

square-roots of Gram’s in

denominators; no Chinese magic

action; iteration for reduction intrinsic

11/21/2011 35


del Aguila-Pittau:

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del Aguila-Pittau:

11/21/2011 37


van Hameren et al.(EPJC41(2005)361):

spinor representation of external

tensor coefficient of massless one-

loop n-point tensor integral to reduce

iteratively rank of that integral for

numerical evaluation using Dirac

matrix methods – square roots of

Gram’s in denominators; no Chinese

magic action

11/21/2011 38


van Hameren et al.:

11/21/2011 39


OPP: See for example,

From integrand N(q)/(D0…Dn-1), expand

N(q) in powers of {Dj} with coefficients

that have a q independent part and a q

dependent part which integrates to

zero with integration measure in d

dimensions for N in 4-dim.

11/21/2011 40


OPP: various methods for adding the rational terms

have been given, including the d-dimensional unitarity

version(Ellis et al., see also arXiv:11054319) --

or taken as given, no Chinese magic action

at loop momentum level to simplify; rational terms

essential whereas we do not need to leave 4-dim and

get the entire contribution directly; inverse Gram’s in

coefficients – numerical stability issues; coefficients solved for by requiring that 4, 3, 2, 1 of the {Dj} vanish

leading to evaluation of N at complex values of q --

issues of Hilbert space extension can obtain, i.e., 3

gluon vertex on-shell is zero in QCD for real momenta,

not so for complex -- we do not have such issues.

q

N q( ) N q( )

11/21/2011 41

Relationship to Other Approaches OPP:

q


BDK(Ann.Phys.322(2007)1587 & refs.

therein): N(q) taken as given,

constructs amplitude from unitarity-

based on-shell recursion relations,

where authors can get both rational

and cut-constructable parts —

No Chinese magic action at loop

variable level to simplify result;

11/21/2011 42

11/21/2011 43


solution of on-shell relations leads to evaluation of N at complex momenta and leads in general to troublesome kinematical factors in the denominators of the representation coefficients for that solution, raising issues of numerical stability. We avoid these latter issues

-- we have a cross-check.

GOLEM(Cullen et al., arXiv:1007.3580 ):

Tensor basis with form factors/OPP. Can in principle eliminate Gram’s as well; no Chinese magic action at loop variable level .

11/21/2011 44


BDK:

GOLEM:

11/21/2011 45

Summary

New method for direct reduction of tensor one(n)-loop integrals with Chinese magic action at loop variable(s) level, with no Gram’s , 4-dimensional loop momenta.

Applications in progress.

Stay tuned!

Documents

Magic Spinor Product Methods in Loop Integralshep.baylor.edu/hatake/HEPseminar/talks/Ward_20111121_Mag...11/21/2011 33 More comparison -- New Trace Over Fermion Lines Method (Glover&Tejeda-Yeomans)