introduction to the soft collinear eﬀective theoryiains/talks/scet-latticeeft-2b.pdf · the only...

Iain StewartMIT

Introduction to the Soft - Collinear Effective Theory

Lecture II

Methods of Effective Field Theory & Lattice Field Theory FGZ-PH Summer School, Munich, Germany

July 2017

Outline (Lecture I)

SCET1 , momentum scales and regions

Wilson lines, W, from off shell propagators

•

•

•

Field power counting in SCET

•

•

• Gauge Symmetry

• Hard-Collinear Factorization

eg. Deep Inelastic Scattering

EFT conceptsIntro to SCETSCET degrees of freedom

�Done on the Board( See separate lecture notes. )

SCET Lagrangian

•

•

•

Outline (Lecture II)

Review from Lecture I

Hard Operator Examples

•

Sudakov Resummation from RGE•

Soft-Collinear Factorization

(& One-loop Matching Example)

• & Factorizatione+e� � dijets

�Done on the Board( See separate lecture notes. )

• Quasi Parton Distribution Function� On board (separate

lecture notes. )

LO SCET Lagrangian:

SCETI summary

usoft & collinear modes⇥n � �

L(0)us = LQCD(qus, A

µus)

I

qus � �3

Aµus � �2

covariant derivatives:

,

iDµus = i�µ + gAµ

us

(A+n , A�n , A⇥n ) � (�2, 1,�)

� pµc

p+

c hard

l2

2

p-

Q

lQ 0

cn

lQ lQ 0

us

n

iDnµ� = i�µ

n� + gA�µn

in·Dn = in · �n + gn·An

L(0)n� = �n

�n · iDus + gn · An + i /Dn

�1

in · Dni /Dn�

� /n

2�n

L(0)ng = L(0)

ng (Dµn�, n·Dn, in·Dus + gn·An)

Properties of

1)

2)

3)

has particles and antiparticles, pair creation & annihilationin/

2⇥(n·p)

n·p + p2�

n·p + i�+

in/

2⇥(�n·p)

n·p + p2�

n·p � i�=

in/

2n·p

n·p n·p + p2⇥ + i�

=in/

2n·p

p2 + i�

all components of Aµn couple to �n

only n·Aus couple at LO

p

qin/

2n·(p + q)

(p + q)2 + i�all components of

p & q appear n

n

n

n

n

p

kusoft

in/

2n·p

n·p n·(p+k) + p2� + i�

=in/

2n·p

n·p n·k + p2 + i�

=in/

21

n·k + i�onshell p2 = 0eikonal

, only depends on n·kus momentum

L(0)n� = �n


�1

in · Dni /Dn�

� /n

2�n

Reparameterization Invariance (RPI)

9

i∂µUc(x) ∼ pµcUc(x) ↔ Aµ

n,q (74)

i∂µUus(x) ∼ pµc Uus(x) ↔ Aµ

us (75)

n n, break Lorentz invariance, restored within collinear cone by RPI, three types

(b) Any choice of the reference light-cone vectors n and n satisfying

n2 = 0 , n2 = 0 , n · n = 2 , (3)

are equally good, and can not change physical predictions.

For type (b) the most general infinitesimal change in n and n which preserves Eq. (3) is alinear combination of

(I)

⇤⇧

⌅nµ ⌅ nµ + �⌅

µ

nµ ⌅ nµ

(II)

⇤⇧

⌅nµ ⌅ nµ

nµ ⌅ nµ + ⌃⌅µ(III)

⇤⇧

⌅nµ ⌅ (1 + �) nµ

nµ ⌅ (1� �) nµ

, (4)

where {�⌅µ , ⇥⌅µ , �} are five infinitesimal parameters, and n ·⌃⌅ = n ·⌃⌅ = n ·�⌅ = n ·�⌅ = 0.

Invariance under subset (I) of these transformations has already been explored in Ref. [15],and used to derive important constraints on the next-to-leading order collinear Lagrangianand heavy-to-light currents. Here we explore the consequences of invariance under the full setof reparameterization transformations and extend the analysis of class (I) transformationsto higher orders in ⇤. In particular we show that the transformations in classes (II) and (III)are necessary to rule out the possibility of additional operators in the lowest order collinearLagrangian that are allowed by power counting and gauge invariance.

As might be expected the collinear reparameterization invariance is a manifestation ofthe Lorentz symmetry that was broken by introducing the vectors n and n. Essentiallyreparameterization invariance restores Lorentz invariance to SCET order by order in ⇤. Thefive parameters in Eq. (4) correspond to the five generators of the Lorentz group which are“broken” by introducing the vectors n and n, namely {nµMµ⇥ , nµMµ⇥}. If the perpendiculardirections are 1, 2 then the five broken generators are Q±

1 = J1±K2, Q±2 = J2±K1, and K3.

The type (I) transformations are equivalent to the combined actions of an infinitesimal boostin the x (y) direction and a rotation around the y (x) axis, such that nµ is left invariantwith generators (Q�

1 , Q+2 ). Type (II) transformations are similar but (Q+

1 , Q�2 ) leave nµ

invariant, while transformation (III) is a boost along the 3 direction (K3).In SCET one introduces three classes of fields: collinear, soft and ultrasoft (usoft), with

momentum scaling as Q(⇤2, 1, ⇤), Q(⇤, ⇤, ⇤) and Q(⇤2, ⇤2, ⇤2), respectively. For our purposesthe interesting fields are those for collinear quarks (⌅n,p), collinear gluons (An,q), and usoftgluons (Au). At tree level the transition from QCD to collinear quark fields can be achievedby a field redefinition [2]

⇧(x) =⌃

p

e�ip·x�1 +

1

n · D D/⌅ n/

2

⇥⌅n,p, (5)

where the two-component collinear quark field ⌅n satisfies [1]

n/n/

4⌅n = ⌅n , n/ ⌅n = 0 . (6)

The covariant derivatives are further decomposed into two parts, Dµ = Dµc + Dµ

u , where Dµc

and Dµu involve collinear and usoft momenta and gauge fields respectively. To distinguish

3

nµ

nµ

nµ

nµ

nµ

nµ

longitudinal boost

unique

∆⊥µ ∼ λ ε

⊥µ ∼ λ

0α ∼ λ

0

Symmetries: Gauge Invariance

L(0)n� = �n


�1

in · Dni /Dn�

� /n

2�n

Together:

L(0)SCETI

= L(0)us +

�

n

�L(0)

n� + L(0)ng

�+ L(0)

Glauber

Full QCDfor qus, Aµ

ussum over distinct RPIequivalence classesn1 · n2 � �2

extra term encodingGlauber gluon exchange,

only factorization violating term

19

a)

qn

n

n

n

qn

n

n

nq

n

n

n

n

qn

n

n

n

n

n

n

n=

n

n

n

n

n

n

n

n=

n

n

n

n

b)

n

n

n

n=

n

n

n

nn nn n

=n

n

n

n

FIG. 4. Tree level matching for the nnnn Glauber operators. In a) we show the four full QCD graphs

with t-channel singularites. The matching results are given by reading down each column. In b) we show

the corresponding Glauber operators for the four operators in SCET with two equivalent notations. The

notation with the dotted line in c) emphasizes the factorized nature of the n and n sectors in the SCET

Glauber operators, which have a 1/P2? between them denoted by the dashed line.

Thus for these tree level 2–2 scattering graphs the Mandelstam invariant t = q2? = �~q 2? < 0.

For this matching calculation there are four relevant QCD tree graphs, shown in Fig. 4a. They

will result in four di↵erent Glauber operators, whose Feynman diagrams for this matching are

represented by Fig. 4c. The matching must be carried out using S-matrix elements for a physical

scattering process, so we take ?-polarization for the external gluon fields. Expanding in � the

results for the top row of diagrams at leading order is

ih

unn/

2TBun

ih�8⇡↵s(µ)�BC

~q 2?

ih

vnn/

2TCvn

i

, (28)

ih

ifBA3A2gµ2µ3? n · p2


~q 2?

ih

vnn/

2TCvn

i

,

ih

unn/

2TBun


~q 2?

ih

ifCA4A1gµ1µ4? n · p1

i

,

ih

ifBA3A2gµ2µ3? n · p2


~q 2?

ih

ifCA4A1gµ1µ4? n · p1

i

.

In writting these results we have written out the collinear quark spinors but left o↵ the collinear

gluon polarization vectors "µ2A2n (p2) etc, for simplicity.

We begin our analysis by discussing the SCETII operators whose tree level matrix elements

reproduce the results in Eq. (28). The four SCETII operators whose matrix elements reproduce

Eq. (28) factorize into collinear and soft operators separated by 1/P2? factors, so we adopt the

1k2�

see arXiv:1601.04695

L(0)Glauber(�n1 , An1 , �n2 , An2 , . . . , qs, As)

Sudakov Logs & RGE (Renormalization Group Equations)UV renormalization in SCET

e+e� � dijets �n�µ��n =

��nWn

��µ�

�W †

n�n

�

8.1 b ! s�, SCET Loops and Divergences 8 WILSON COEFFICIENTS AND HARD DYNAMICS

Next consider the ultrasoft loops in SCET. In Feynman gauge the ultrasoft wavefunction renormal-ization of the collinear quark vanishes, since the couplings are both proportional to nµ, and n2 = 0. Theultrasoft wavefunction renormalization of the heavy quark is just the HQET wavefunction renormalization.We summarize these two results as:

Zus⇠n

/ nµnµ = 0 , Zushv

= 1 +↵sCF

4⇡

⇣2

✏� 2

✏IR

⌘

. (8.9)

We can already note that the 1/✏IR pole in Zushv

matches up with the IR pole in Z b

in full QCD (and this isthe only IR divergence that we are regulating with dimensional regularization). In addition to wavefunctionrenormalization there is an ultrasoft vertex diagram for the SCET current. Using the on-shell conditionv · pb = 0 for the incoming b-quark, and the SCET propagator from Eq. (5.43) for a line with injectedultrasoft momentum, we have

= V 1us = (ig)2(�i)CF un�uv

Z

d�dk µ2✏◆✏ n · v(v · k + i0)(n · k + p2/n · p+ i0)(k2 + i0)

V 1us = �↵sCF

4⇡

1

✏2+

2

✏ln⇣ µn · p�p2�i0

⌘

+ 2 ln2⇣ µn · p�p2�i0

⌘

+3⇡2

4

�

V 0scet , (8.10)

where the tree level SCET amplitude is

V 0scet = un�uv , (8.11)

and ◆✏ = (4⇡)�✏e✏�E ensures that the scale µ has the appropriate normalization for the MS scheme. Notethat this graph is independent of the current’s Dirac structure �. On the heavy quark side the heavy-quark propagator gives a Pv = (1 + v/)/2, but this commutes with the HQET vertex Feynman rule andhence yields a projector on the HQET spinor, Pvuv = uv. On the light quark side the propagator givesa n//2 and the vertex gives a n//2 to yield the projector Pn = (n/n/)/4 acting on the light-quark spinor,Pnun = un. Hence whatever � is inserted at the current vertex is also the Dirac structure that appearsbetween spinors in the answer for the loop graph. For this heavy-to-light current this feature is actuallytrue for all loop diagrams in SCET, the spin structure of the current is preserved by loops diagrams inthe EFT. For ultrasoft diagrams it happens by a simple generalization of the arguments above, while forcollinear diagrams the interactions only appear on the collinear quark side of the �, so we just need toknow that they do not induce additional Dirac matrices. (This is ensured by chirality conservation in theEFT.)

Lets finally consider the one loop diagrams with a collinear gluon. There is no wavefunction renormal-ization diagram for the heavy quark, since the collinear gluon does not couple to it. There is a wavefunctionrenormalization graph for the light-collinear quark

= . . . =n/

2

p2

n · pCF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

, so Z⇠n

= 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

.

(8.12)

We have not written out the SCET loop integrand, but it follows in a straightforward manner from usingthe collinear quark and gluon propagators and vertex Feynman rules from Fig. (6). Note that the result forZ⇠

n

is the same as the full theory Z . This occurs because for the wavefunction graph there is no connection

57

8.2 e+e� ! 2-jets, SCET Loops 8 WILSON COEFFICIENTS AND HARD DYNAMICS

As before, we next consider the loops in SCET. The wave function renormalization for the collinearquark is the same as in the previous section, and we find

Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)

The tree level amplitude in SCET is V 0scet = un(pq)�i vn(pq), and to leading order V 0

qcd = V 0scet. The

ultrasoft vertex graph in SCET involves an exchange between the n-collinear and n-collinear quarks,

and is given by

V 1usoft = µ2✏◆✏

Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2

n · pqn · pq n · k + p2q

�iin/

2

�n · pqn · pq n · k + p2q

⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)

There are two possible collinear vertex graphs which involve a contraction between the Wn[n ·An] Wilsonline and a n-collinear quark, and another between the Wn[n ·An] Wilson line and the n-collinear quark

For the first diagram, we find

V 1coll = µ2✏◆✏

Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)

n · k (p+ k)2 k2un�ivn

=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)

One can easily show that the second collinear vertex diagram gives the same result as the first diagram.Furthermore the collinear integral here is identical to the one for b ! s� in Eq. (8.14). The result inEq. (8.36) is for the naive integrand, since it does not include the 0-bin subtraction contribution. But the

63

= same withp2 � p2



Zus⇠n


= 1 +↵sCF

4⇡

⇣2

✏� 2

✏IR

⌘

. (8.9)





Z


V 1us = �↵sCF

4⇡

1

✏2+

2

✏ln⇣ µn · p�p2�i0

⌘

+ 2 ln2⇣ µn · p�p2�i0

⌘

+3⇡2

4

�

V 0scet , (8.10)


V 0scet = un�uv , (8.11)



= . . . =n/

2

p2

n · pCF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

, so Z⇠n

= 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

.

(8.12)


n


57

� = ��sCF

4�

1�

= 0

eg.



Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)


qcd = V 0scet. The


and is given by


Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2


�iin/

2


⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)




Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)


=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)


63

�sCF

4�

�2�2

+2�� 2

�ln

��p2

µ2

�+ . . .

�=

�ddk

(2�)d

n · (k + p)n · k (p + k)2k2

- (0-bin)



Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)


qcd = V 0scet. The


and is given by


Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2


�iin/

2


⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)




Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)


=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)


63

�sCF

4�

�� 2

�2+

2�

ln�

(�p2)(�p2)(�Q2)µ2

�+ . . .

�=

�ddk

(2�)d

n · n�n · k + p2

Q

��n · k + p2

Q

�k2

sum =�sCF

4�

�2�2

+2�

ln� µ2

�Q2 � i0

�+

3�

+ . . .

�

( Feynman gauge, UV: , IR: ) d = 4� 2� p2 �= 0, p2 �= 0

n nnn

n n

us

us

n

sum =�sCF

4�

�2�2

+2�

ln� µ2

�Q2 � i0

�+

3�

+ . . .

�

counterterm (ZC � 1)� =�sCF

4�

�� 2

�2� 2

�ln

� µ2

�Q2 � i0

�� 3

�+ . . .

�MS

Cbare = ZC C(µ)

0 = µd

dµCbare =

�µ

d

dµZC

�C(µ) + ZC

�µ

d

dµC(µ)

�

RGE:

µd

dµC(µ) = �C C(µ)

= ��s(µ)4�

�4CF ln

µ2

�Q2+ 6CF

�finite

�C =�� Z�1

C

�µ

d

dµZC = (�1)

CF

4�

�(�2� �s)

��2�2� 2

�ln

µ2

�Q2� 3

�

�+ �s

��4�

��

µd

dµ�s = �2� �s + . . .

sum =�sCF

4�

�2�2

+2�

ln� µ2

�Q2 � i0

�+

3�

+ . . .

�

counterterm (ZC � 1)� =�sCF

4�

�� 2

�2� 2

�ln

� µ2

�Q2 � i0

�� 3

�+ . . .

�MS

Cbare = ZC C(µ)

RGE: square the amplitude: H =��C(µ)

��2

H(Q,µ1) = H(Q,µ0) exp��# �s ln2 µ1

Q+ . . .

�

H(Q,µ1) = H(Q,µ0) exp��#

1�s(µ0)

f��s(µ1)

�s(µ0)

�+ . . .

�

Complete Solutionderived in

Homework

frozencoupling

runningcoupling

Sudakov Form Factor�n�µ

��n restricts radiation, Sudakov = no emission probability

µd

dµH(Q,µ) =

��C + ��C

�H(Q,µ) = ��s(µ)

2�

�8CF ln

µ

Q+ 6CF

�H(Q,µ)Q

µ0

µ1

leading log (LL)term

� �

needed at NLL

One-Loop Matching CalculationQCD

SCET

LQCD + J = ��µ�

L(0)SCET + C �n�µ

��n find C at O(�s)�1-loop ren. QCD

��

�1-loop ren. SCET

�= C1loop��n�µ

��n�tree

IR: p2 = p2 �= 0

both theories

One-Loop Matching CalculationQCD

SCET

LQCD + J = ��µ�

L(0)SCET + C �n�µ

��n find C at O(�s)�1-loop ren. QCD

��

�1-loop ren. SCET

�= C1loop��n�µ

��n�tree

QCD +

8 WILSON COEFFICIENTS AND HARD DYNAMICS

8 Wilson Coe�cients and Hard Dynamics

We now turn to the dynamics of SCET at one loop. An interesting aspect of loops in the e↵ective theory isthat often a full QCD loop graph has more than one counterpart with similar topology in SCET. We willcompare the SCET one loop calculation for a single hard interaction current with the one loop calculationin QCD. Our goal is to understand the IR and UV divergences in SCET and the corresponding logarithms,as well as understanding how the terms not associated to divergences are treated.

In our analysis we will use the same regulator for infrared divergences, and show that the IR divergencesin QCD and SCET exactly agree, which is a validation check on the EFT. The di↵erence determinesthe Wilson coe�cient for the SCET operator that encodes the hard dynamics. This matching result isindependent of the choice of infrared regulator as long as the same regulator is used in the full and e↵ectivetheories. Finally, the SCET calculation contains additional UV divergences, beyond those in full QCD,and the renormalization and anomalous dimension determined from these divergences will sum up doubleSudakov logarithms.

We will give two examples of matching QCD onto SCET, the b ! s� transition, and e+e� ! 2-jets. The first example has the advantage of involving only one collinear sector, but the disadvantageof requiring some familiarity with Heavy Quark E↵ective theory for the treatment of the b quark andinvolving contributions from two Dirac structures. The second example only involves jets with a singleDirac structure, but has two collinear sectors. In both cases we will use Feynman gauge for all gluons, anddimensional regularization with d = 4� 2✏ for all UV divergences (denoting them as 1/✏). To regulate theIR divergences we will take the strange quark o↵shell, p2 6= 0. For IR divergences associated purely withthe heavy quark we will use dimensional regularization (denoting them 1/✏IR to distinguish from the UVdivergences).

8.1 b ! s�, SCET Loops and Divergences

As a 1-loop example consider the heavy-to-light currents for b ! s�. Although there are several operatorsin the full electroweak Hamiltonian, for simplicity we will just consider the dominant dipole operatorJQCDµ⌫ Fµ⌫ where Fµ⌫ is the photon field strength and the quark tensor current is

JQCD = s�b , � = �µ⌫PR . (8.1)

In SCET the corresponding current (for the original Lagrangian, prior to making the Yn field redefinition)was

JSCET = (⇠nW )�hvC�

v · n P†�=

Z

d! C(!) �n,!�hv . (8.2)

In general because of the presense of the vectors vµ and nµ there can be a larger basis of Dirac structures� for the SCET current (we will see below that at one-loop there are in fact two non-zero structures forthe SCET tensor current). Note that the factor of v · n makes it clear that the current preserves type-IIIRPI. We will set v · n = 1 in the following.

Together with the QCD and (leading order) SCET Lagrangians, we can carry out loop calculations withthese two currents. First lets consider loop corrections in QCD. We have a wavefunction renormalizationgraph for the heavy quark denoted b, and one for the massless (strange) quark denoted q:

b q

55

�


back-to-back n and n directions

JSCET = (⇠nWn)�iC�

P†n,Pn, µ

�

(W †n⇠n) =

Z

d! d!0 C(!,!0) �n,!0�i �n,! . (8.27)

By reparametrization invariance of type-III the dependence on the label operators can only be in thecombination !!0 inside C, so

C�

!,!0) = C�

!!0� . (8.28)

Finally in the CM frame momentum conservation fixes ! = !0 = Q, the CM energy of the e+e� pair, sowe can write

JSCET = C(Q2) (⇠nWn)�i (W†n⇠n) , (8.29)

and the matching calculation in this section will determine the renormalized MSWilson coe�cient C(Q2, µ2).In this case there is only one relevant Dirac structure �i in SCET for each of the vector and axial-vectorcurrents.

We again begin by calculating the full theory diagrams. As in the case of B ! Xs� we need the wavefunction contributions for the light quarks, in this case one for the quark and one for the anti-quark. Bothwave function contributions are the same as the results obtained before

Z = 1� ↵sCF

4⇡

1

✏� ln

�p2

µ2+ 1

�

. (8.30)

The remaining vertex graph can again be calculated in a straightforward manner. At tree level we find

V 0qcd = u(pn)�ivn(pn) (8.31)

while the one loop vertex diagram

pq

pq

gives

V 1qcd = µ2✏◆✏

Z

ddk

(2⇡)dig u(pq)�

↵TA i(p/q + k/)

(pq + k)2�i

�i(p/q + k/)

(pq + k)2ig�↵T

A v(pq)�i

k2

= ig2CF µ2✏Z

ddk

(2⇡)du(pq)

�↵ (p/q + k/)�i (p/q + k/) �↵(pq + k)2 (pq + k)2 k2

v(pq)

=↵sCF

4⇡

1

✏� 2 ln2

p2

Q2� 4 ln

p2

Q2� ln

(�Q2 � i0)

µ2� 2⇡2

3

�

u(pq)�i v(pq) . (8.32)

Here ◆✏ = (4⇡)�✏e✏�E ensures that the scale µ has the appropriate normalization for the MS scheme. Addingthe QCD diagrams we find

QCD Sum = V 1qcd + 2

h1

2(Z � 1)

i

V 0qcd

=↵sCF

4⇡

�2 ln2p2

Q2� 3 ln

p2

Q2� 1� 2⇡2

3

�

u(pq)�i v(pq) . (8.33)

62SCET



Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)


qcd = V 0scet. The


and is given by


Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2


�iin/

2


⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)




Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)


=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)


63



Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)


qcd = V 0scet. The


and is given by


Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2


�iin/

2


⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)




Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)


=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)


63



Zus⇠n


= 1 +↵sCF

4⇡

⇣2

✏� 2

✏IR

⌘

. (8.9)





Z


V 1us = �↵sCF

4⇡

1

✏2+

2

✏ln

⇣ µn · p�p2�i0

⌘

+ 2 ln2⇣ µn · p�p2�i0

⌘

+3⇡2

4

�

V 0scet , (8.10)


V 0scet = un�uv , (8.11)



= . . . =n/

2

p2

n · pCF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

, so Z⇠n

= 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

.

(8.12)


n


57

�



Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)


qcd = V 0scet. The


and is given by


Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2


�iin/

2


⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)




Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)


=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)


63

+ + + + �(ZC � 1)

=�sCF

4�

�2 ln2 µ2

�p2+ 3 ln

µ2

�p2� ln2

�µ2Q2

�p4

�+ 7� 5�2

6

�

=�sCF

4�

�ln2 µ2

�Q2� 2 ln2 p2

Q2� 3 ln

p2

Q2+ 3 ln

µ2

�Q2+ 7� 5�2

6

�

collinearsoft

IR divergences match

IR: p2 = p2 �= 0

both theories

One-Loop Matching CalculationQCD� SCET =

�sCF

4�

�� ln2 µ2

�Q2� 3 ln

µ2

�Q2� 8 +

�2

6

�

C(Q,µ) = 1 +�s(µ)CF

4�

�� ln2 µ2

�Q2� 3 ln

µ2

�Q2� 8 +

�2

6

�

One-Loop Matching CalculationQCD� SCET =

�sCF

4�

�� ln2 µ2

�Q2� 3 ln

µ2

�Q2� 8 +

�2

6

�

C(Q,µ) = 1 +�s(µ)CF

4�

�� ln2 µ2

�Q2� 3 ln

µ2

�Q2� 8 +

�2

6

�

Once we know how this works, there is a much easier way to get this answer. Result for C is independent of our choice of IR regulator. Use dim.reg. for IR too.

+

8 WILSON COEFFICIENTS AND HARD DYNAMICS

8 Wilson Coe�cients and Hard Dynamics

We now turn to the dynamics of SCET at one loop. An interesting aspect of loops in the e↵ective theory isthat often a full QCD loop graph has more than one counterpart with similar topology in SCET. We willcompare the SCET one loop calculation for a single hard interaction current with the one loop calculationin QCD. Our goal is to understand the IR and UV divergences in SCET and the corresponding logarithms,as well as understanding how the terms not associated to divergences are treated.

In our analysis we will use the same regulator for infrared divergences, and show that the IR divergencesin QCD and SCET exactly agree, which is a validation check on the EFT. The di↵erence determinesthe Wilson coe�cient for the SCET operator that encodes the hard dynamics. This matching result isindependent of the choice of infrared regulator as long as the same regulator is used in the full and e↵ectivetheories. Finally, the SCET calculation contains additional UV divergences, beyond those in full QCD,and the renormalization and anomalous dimension determined from these divergences will sum up doubleSudakov logarithms.

We will give two examples of matching QCD onto SCET, the b ! s� transition, and e+e� ! 2-jets. The first example has the advantage of involving only one collinear sector, but the disadvantageof requiring some familiarity with Heavy Quark E↵ective theory for the treatment of the b quark andinvolving contributions from two Dirac structures. The second example only involves jets with a singleDirac structure, but has two collinear sectors. In both cases we will use Feynman gauge for all gluons, anddimensional regularization with d = 4� 2✏ for all UV divergences (denoting them as 1/✏). To regulate theIR divergences we will take the strange quark o↵shell, p2 6= 0. For IR divergences associated purely withthe heavy quark we will use dimensional regularization (denoting them 1/✏IR to distinguish from the UVdivergences).

8.1 b ! s�, SCET Loops and Divergences

As a 1-loop example consider the heavy-to-light currents for b ! s�. Although there are several operatorsin the full electroweak Hamiltonian, for simplicity we will just consider the dominant dipole operatorJQCDµ⌫ Fµ⌫ where Fµ⌫ is the photon field strength and the quark tensor current is

JQCD = s�b , � = �µ⌫PR . (8.1)

In SCET the corresponding current (for the original Lagrangian, prior to making the Yn field redefinition)was

JSCET = (⇠nW )�hvC�

v · n P†�=

Z

d! C(!) �n,!�hv . (8.2)

In general because of the presense of the vectors vµ and nµ there can be a larger basis of Dirac structures� for the SCET current (we will see below that at one-loop there are in fact two non-zero structures forthe SCET tensor current). Note that the factor of v · n makes it clear that the current preserves type-IIIRPI. We will set v · n = 1 in the following.

Together with the QCD and (leading order) SCET Lagrangians, we can carry out loop calculations withthese two currents. First lets consider loop corrections in QCD. We have a wavefunction renormalizationgraph for the heavy quark denoted b, and one for the massless (strange) quark denoted q:

b q

55

� �sCF

4�

�� 2

�2IR� 2

�IRln

µ2

�Q2� 3

�IR� ln2 µ2

�Q2� 3 ln

µ2

�Q2� 8 +

�2

6

�=



Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)


qcd = V 0scet. The


and is given by


Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2


�iin/

2


⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)




Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)


=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)


63



Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)


qcd = V 0scet. The


and is given by


Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2


�iin/

2


⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)




Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)


=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)


63



Zus⇠n


= 1 +↵sCF

4⇡

⇣2

✏� 2

✏IR

⌘

. (8.9)





Z


V 1us = �↵sCF

4⇡

1

✏2+

2

✏ln

⇣ µn · p�p2�i0

⌘

+ 2 ln2⇣ µn · p�p2�i0

⌘

+3⇡2

4

�

V 0scet , (8.10)


V 0scet = un�uv , (8.11)



= . . . =n/

2

p2

n · pCF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

, so Z⇠n

= 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

.

(8.12)


n


57

�



Zus⇠ = 0 , Z⇠ = 1� CF↵s

4⇡

⇣1

✏� ln

�p2

µ2+ 1

⌘

. (8.34)


qcd = V 0scet. The


and is given by


Z

ddk

(2⇡)dun

⇣

ign/

2n↵TA

⌘ in/

2


�iin/

2


⇣

ign/

2n↵T

A⌘

vn�i

k2

= ig2CFµ2✏◆✏

⇣

unn/n/

4�i

n/n/

4vn

⌘

Z

ddk

(2⇡)dn · n

⇣

n · k +p2q

n·pq

⌘⇣

n · k +p2q

n·pq

⌘

k2

=↵sCF

4⇡

� 2

✏2+

2

✏ln

�p4

µ2Q2� ln2

�p4

µ2Q2� ⇡2

2

�

un(pq)�ivn(pq) . (8.35)




Z

ddk

(2⇡)dig un

n↵ +�?p/?n · p +

(p/? + k/?)�↵?n · (p+ k)

� p/?(p/? + k/?)n · pn · (p+ k)

�

n/

2TA

⇥ in/

2

n · (p+ k)

(p+ k)2

⇣

�gn↵

n · kTA⌘ �i

k2�i vn

= �ig2CFµ2✏◆✏

Z

ddk

(2⇡)d(n · n) n · (p+ k)


=↵sCF

4⇡

2

✏2+

2

✏� 2

✏ln

�p2

µ2+ ln2

�p2

µ2� 2 ln

�p2

µ2+ 4� ⇡2

6

�

un(pq)�i vn(pq) . (8.36)


63

+ + + � 1�2UV

� 1�2IR

,1

�UV� 1

�IR

vanish for�UV = �IR

�(ZC � 1) =�sCF

4�

�� 2

�2UV

� 2�UV

ln� µ2

�Q2 � i0

�� 3

�UV+ . . .

�

QCD� SCET =�sCF

4�

�� ln2 µ2

�Q2� 3 ln

µ2

�Q2� 8 +

�2

6

�= same as above = IR finite part of QCD

calculation with this regulator

Ultrasoft - Collinear Factorization

Multipole Expansion: L(0)c = ªn

n

n · iDus + gn · An + i /Dc?

1in · Dc

i /Dc?

o /n

2ªn

usoft gluons have eikonal Feynman rules and induce eikonal propagators

gives

Factorization of Usoft Gluons

Consider the following field redefinitions in SCET

�n,p = Yn �(0)n,p , An,q = Yn A(0)

n,q Y †n

where Yn = Pexp⌃ig

⌅ x�⇤ ds n·Aus(ns)

⇥, n·DYn = 0, and Y †

n Yn = 1

Find

• Lq = �n,p�⌥in·D + . . .

⇧�n,p =� �(0)

n,p�

⇤in·⇧ + . . .

��(0)n,p

• W = YnW (0)Y †n

• L(�n,p, Aµn,q, n·Aus) = L(�(0)

n,p, A(0)µn,q , 0)

Moves all usoft gluons to operators, simplifies cancellations

eg1. J = �(0)n W (0) ⌃ Y †

n hv

eg2. J = �nW ⌃ W †�n = �(0)n W (0) ⌃ W (0)†�(0)

n

Iain Stewart – p.15

k

in·k+iϵ k

i−n·k+iϵ k

i−n·k−iϵ k

in·k−iϵ

(Y+ ξ+n ) (ξ+

n Y†+) (ξ−n Y

†−) (Y− ξ−n )

FIG. 1: Eikonal iϵ prescriptions for incoming/outgoing quarks and antiquarks and the result thatreproduces this with an ultrasoft Wilson line and sterile quark field.

Since the dependence on s0 sometimes causes confusion, we explore some of the subtletiesin this section, in particular, why it is important to remember that factors of Y , Y † canalso be induced in the interpolating fields for incoming and outgoing collinear states, andwhy a common choice for s0 = s †

0 is sufficient to properly reproduce the iϵ prescription inperturbative computations. In many processes (examples being color allowed B → Dπ andB → Xsγ) the s0 dependence of the Wilson lines cancels and the following considerationsare not crucial. In other processes, however, the path for the Wilson line is important for thefinal result, particularly when these Wilson lines do not entirely cancel. An example of thisis jet event shapes as discussed in Refs. [28–30]. See also the discussion of path dependencein eikonal lines in Refs. [31–37].

First consider the perturbative computation of attachments of usoft gluons to incomingand outgoing quark and antiquark lines. The results for the eikonal factors for one gluonare summarized in Fig. 1, and can be computed directly with the SCET collinear quarkLagrangian (or from an appropriate limit of the QCD propagator). These attachments seemto force one to make a particular choice for s0 and s0, see for example the recent detailedstudy in Ref. [30]. In our notation it is straightforward to show that this choice correspondsto

s0 = −∞ sign(P) , s0 = +∞ sign(P†) ,

!

P=P′=P , for P , P† > 0

P=P′=P , for P , P† < 0

. (21)

To see this take a quark with label n·p > 0 and an antiquark with label n·p′ < 0, and notethat

Y ξn,p = P exp"

ig

# 0

−∞

ds n·Aus(xµs )

$

ξ+n,p = P exp

"

ig

# 0

−∞

ds n·Aus(xµs )

$

ξ+n,p ≡ Y+ξ+

n,p , (22)

ξn,pY†= ξ+

n,pP′exp

"

−ig

# 0

∞

ds n·Aus(xµs )

$

= ξ+n,pP exp

"

ig

# ∞

0

ds n·Aus(xµs )

$

≡ ξ+n,pY

†+ ,

Y ξn,p′ = P exp"

ig

# 0

∞

ds n·Aus(xµs )

$

ξ−n,p′ = P exp"

−ig

# ∞

0

ds n·Aus(xµs )

$

ξ−n,p′ ≡ Y−ξ−n,p′ ,

ξn,p′Y†= ξ−n,p′P

′exp

"

−ig

# 0

−∞

ds n·Aus(xµs )

$

= ξ−n,p′P exp"

−ig

# 0

−∞

ds n·Aus(xµs )

$

≡ ξ−n,p′Y†− .

This is in agreement with the Y = Y−, Y † = Y †−, Y = Y+, Y † = Y †

+ used in [30] for theproduction and annihilation of antiparticles and the annihilation and production of parti-cles respectively. The results in Eq. (22) reproduce the natural choice of having incomingquarks/antiquarks enter from −∞, while outgoing quarks/antiquarks extend out to +∞.

7

k

in·k+iϵ k

i−n·k+iϵ k

i−n·k−iϵ k

in·k−iϵ

(Y+ ξ+n ) (ξ+

n Y†+) (ξ−n Y

†−) (Y− ξ−n )





s0 = −∞ sign(P) , s0 = +∞ sign(P†) ,

!

P=P′=P , for P , P† > 0

P=P′=P , for P , P† < 0

. (21)


Y ξn,p = P exp"

ig

# 0

−∞

ds n·Aus(xµs )

$

ξ+n,p = P exp

"

ig

# 0

−∞

ds n·Aus(xµs )

$

ξ+n,p ≡ Y+ξ+

n,p , (22)

ξn,pY†= ξ+

n,pP′exp

"

−ig

# 0

∞

ds n·Aus(xµs )

$

= ξ+n,pP exp

"

ig

# ∞

0

ds n·Aus(xµs )

$

≡ ξ+n,pY

†+ ,

Y ξn,p′ = P exp"

ig

# 0

∞

ds n·Aus(xµs )

$


−ig

# ∞

0

ds n·Aus(xµs )

$

ξ−n,p′ ≡ Y−ξ−n,p′ ,


′exp

"

−ig

# 0

−∞

ds n·Aus(xµs )

$

= ξ−n,p′P exp"

−ig

# 0

−∞

ds n·Aus(xµs )

$

≡ ξ−n,p′Y†− .



7

k

in·k+iϵ k

i−n·k+iϵ k

i−n·k−iϵ k

in·k−iϵ

(Y+ ξ+n ) (ξ+

n Y†+) (ξ−n Y

†−) (Y− ξ−n )





s0 = −∞ sign(P) , s0 = +∞ sign(P†) ,

!

P=P′=P , for P , P† > 0

P=P′=P , for P , P† < 0

. (21)


Y ξn,p = P exp"

ig

# 0

−∞

ds n·Aus(xµs )

$

ξ+n,p = P exp

"

ig

# 0

−∞

ds n·Aus(xµs )

$

ξ+n,p ≡ Y+ξ+

n,p , (22)

ξn,pY†= ξ+

n,pP′exp

"

−ig

# 0

∞

ds n·Aus(xµs )

$

= ξ+n,pP exp

"

ig

# ∞

0

ds n·Aus(xµs )

$

≡ ξ+n,pY

†+ ,

Y ξn,p′ = P exp"

ig

# 0

∞

ds n·Aus(xµs )

$


−ig

# ∞

0

ds n·Aus(xµs )

$

ξ−n,p′ ≡ Y−ξ−n,p′ ,


′exp

"

−ig

# 0

−∞

ds n·Aus(xµs )

$

= ξ−n,p′P exp"

−ig

# 0

−∞

ds n·Aus(xµs )

$

≡ ξ−n,p′Y†− .



7

k

in·k+iϵ k

i−n·k+iϵ k

i−n·k−iϵ k

in·k−iϵ

(Y+ ξ+n ) (ξ+

n Y†+) (ξ−n Y

†−) (Y− ξ−n )





s0 = −∞ sign(P) , s0 = +∞ sign(P†) ,

!

P=P′=P , for P , P† > 0

P=P′=P , for P , P† < 0

. (21)


Y ξn,p = P exp"

ig

# 0

−∞

ds n·Aus(xµs )

$

ξ+n,p = P exp

"

ig

# 0

−∞

ds n·Aus(xµs )

$

ξ+n,p ≡ Y+ξ+

n,p , (22)

ξn,pY†= ξ+

n,pP′exp

"

−ig

# 0

∞

ds n·Aus(xµs )

$

= ξ+n,pP exp

"

ig

# ∞

0

ds n·Aus(xµs )

$

≡ ξ+n,pY

†+ ,

Y ξn,p′ = P exp"

ig

# 0

∞

ds n·Aus(xµs )

$


−ig

# ∞

0

ds n·Aus(xµs )

$

ξ−n,p′ ≡ Y−ξ−n,p′ ,


′exp

"

−ig

# 0

−∞

ds n·Aus(xµs )

$

= ξ−n,p′P exp"

−ig

# 0

−∞

ds n·Aus(xµs )

$

≡ ξ−n,p′Y†− .



7

,

Field Redefinition:

�n � Yn�n An � YnAnY †n Yn(x) = P exp

�ig

� 0

��ds n·Aus(x+ns)

�

L(0)n� = �n

�n · iDus + . . .

� /n

2�n =� �n

�n · iDn + i /Dn�

1in · Dn

i /Dn�

� /n

2�n

similar for L(0)ng

n·DusYn =0, Y †n Yn =1

eg2.color transparency

Field Theory gives the same results pre- and post- field redefinition, but the organization is different

Ultrasoft - Collinear Factorization:

eg1.

usoft-collinear factorization is simple in SCET

�n�µ��n =� �n

�Y †

n Yn

��µ��n

�n/n

2�n =� �n

�Y †

n Yn

� /n

2�n = �n

/n

2�n

note: not upset bysince ultrasoft gluons carry no momenta

�(� � in · �n)in · �n � �0

�n � Yn�n

also Wn � YnWnY †n

eg. e+e−

→ 2 jetsdd

usoft particles

n-collinear jet

n-collinear jet

dσ

de=

1

Q2

!

X

!Lµν

"

0#

#J†ν(0)#

#X$"

X#

#Jµ(0)#

#0$

δ(e − e(X))δ4(q − pX)

event shape in two jet region

J (0) =

!dωdω C(ω, ω) χn,ωΓχn,ω

SCETI

|X⟩ = |XnXnXus⟩

=

!dωdω C(ω, ω) χn,ω Y †

nΓYn χn,ω

�n,� = �(� � in · �n)�n

dσ

de=

1

Q2

!

Xus,Xn,Xn

!Lµν

"

[dωi]C(ω, ω)C(ω′, ω′)#

0$

$(Y †n ΓYn)

$

$Xus

%#

Xus

$

$(Y †n ΓYn)

$

$0%

#

0$

$χn,ω′

$

$Xn

%#

Xn

$

$χn,ω

$

$0%#

0$

$χn,ω′

$

$Xn

%#

Xn

$

$χn,ω

$

$0%

δ(e − e(X))δ4(q − pX)

should specify “e” to go further. One example is thrust:

p+

cn

u

hard

l2

2

p-

Q

lQ 0

cn

lQ lQ 0

SCETI

hard perturbativecorrections

perturbativejet functions

soft function

Homework:Compute the jet functionat one-loop

� = 1� T � 1

d�

d�= �0

��C(Q,µ)��2

�d�+ d��ds ds� J(s�Q�+, µ)J(s� �Q��, µ)S(��, �+, µ)�

�� s + s�

Q2

�

•

•

sum large terms with RGE of C, J, S

�s ln2 �

dominant nonperturbative hadronizationcorrections contained in S:

S(�+, ��, µ) =�

dk dk� Spert(�+ � k, �� k�, µ) F (k, k�)

J1

2

3

−

+

J

J

p

p

Introduction More Introduction Fixed Order Resummation Monte Carlo Summary

Particle Physics: Physics at Shortest Distances

ud

u

m 110510101015 10�5 10�10 10�15

LHC

Frank Tackmann (MIT) Better Theory Predictions for the LHC 2010-11-22 1 / 34

time

Non-perturbative Factorization:

parton distributions

d� = fafb � � � F

perturbative partonic

cross section

hadronization(eg. frag. functions)

proton-proton collision

pp� Higgs + anything

d� =�

dY�

i,j

�d�a

�a

d�b

�bfi(�a, µ)fj(�b, µ) H incl

ij

�mHeY

Ecm�a,mHe�Y

Ecm�b,mH , µ

�eg. Inclusive Higgs production

J1

2

3

−

+

J

J

p

p

µp � �QCD

µJ � mJ

µS

µJ , µB

µH

µp

E

µS � psoft

µB

Perturbative Factorization:

SCET

QCD

µH � Q

µB µH µJ µS

hard jet pert. soft beam

�fact = IaIb �H ��

iJi � S

for multi-scale problems

Introduction More Introduction Fixed Order Resummation Monte Carlo Summary

Particle Physics: Physics at Shortest Distances

ud

u

m 110510101015 10�5 10�10 10�15

LHC

Frank Tackmann (MIT) Better Theory Predictions for the LHC 2010-11-22 1 / 34

introduction to the soft collinear eﬀective theoryiains/talks/scet-latticeeft-2b.pdf · the only...

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