Effective Field Theory of Forward Scattering and Factorization - - PowerPoint PPT Presentation
Effective Field Theory of Forward Scattering and Factorization Violation In collaboration with Iain Stewart SCET 2015 Santa Fe Total Scattering Cross Sections in QFT Consider the total scattering cross section for neutral particles Z d
In collaboration with Iain Stewart
SCET 2015 Santa Fe
Consider the total scattering cross section for neutral particles
σ(s) = Z dσ dt dt This integral is dominated by the region t ⌧ s
some measure of traverse momentum transfer at intermediate stage of calculation
t
Break up integral into regions with distinct power counting parameters t ⌧ s λ ≡ t/s s ⇠ t Λ λ ≡ Λ/(t, s) SCET-I,II SCETII-like 1: 2:
Consider region 1: There must exist some underlying hard event which must be integrated out generating some higher dimensional external operator
S ∼ C(s, t) Z d4x¯ ξnξ¯
n ¯
ξn1ξn2 ∼ λ4
Region 2: No underlying hard interaction, at the scale generate the interaction
t
(λn, λm, λ) : (m + n) ≥ 3
“Glauber” mode S ∼ Z d4x ¯ ξnξ¯
n
1 t ¯ ξnξ¯
n ∼ λ−2λ2λ−2λ2 ∼ 1
The Glauber mode
“V ” ∼ δ(x+)δ(x−) log(x2
⊥)
Strong analogy with NRQCD, Coulomb kernel is dressed by soft gluons
Note that while this operator is at the heart of region 2, it also exists at leading
seed of doubt on factorization proofs.
Note: no hard interactions to all orders in perturbation theory. The scale plays no dynamical role. s = (p1 + p2)2 Consider the dressing of the Glauber kernel by soft gluons M ∼ f(s/t, t/m2) t The scale s can only show up in logs: RG can not hope to capture the logs, need RRG log(p+/ν) + log(p−/ν) + log(t/ν2) = log(s/t) F(s/t) ∼ (s/t)γ Regge behaviour
k+ k− Q λQ λ2Q λ2Q λQ Q n-coll. ¯ n-coll. soft
Matching onto the action
18
a)
q n n n n
q n n n n q n n n n
q n 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 n
n n n n
=
n n n n
Oqq
ns¯ n = OqB n
1 P2
⊥
OBC
s
1 P2
⊥
OqC
¯ n ,
Ogq
ns¯ n = OgB n
1 P2
⊥
OBC
s
1 P2
⊥
OqC
¯ n ,
Oqg
ns¯ n = OqB n
1 P2
⊥
OBC
s
1 P2
⊥
OgC
¯ n ,
Ogg
ns¯ n = OgB n
1 P2
⊥
OBC
s
1 P2
⊥
OgC
¯ n .
Operator basis: OBC
s
= 8παsP2
⊥δBC + ....
OqB
n
= χn,ωT B ¯ n / 2 χn,ω , OgB
n
= i 2fBCDBC
n?µ,ω ¯
n · (P+P†)BDµ
n?,ω .
Must allow for soft emission
Matching Soft-collinear Operators
22
a)
q n s n s
q n s n s q n s n s
q n s n s
b)
n s n s n s n s n s n s s s n n
P⊥ Oqq
ns = OqB n
1 P2
⊥
OqnB
s
, Oqg
ns = OqB n
1 P2
⊥
OgnB
s
, Ogq
ns = OgB n
1 P2
⊥
OqnB
s
, Ogg
ns = OgB n
1 P2
⊥
OgnB
s
.
Matching is identical to the collinear-collinear case
OqnB
s
= 8⇡↵s ⇣ ¯ n
S T B n
/ 2 n
S
⌘ , OgnB
s
= 8⇡↵s ⇣ i 2fBCDBnC
S⊥µ n · (P+P†)BnDµ S⊥
⌘ .
LII(0)
G
= X
n,¯ n
X
i,j=q,g
Oij
ns¯ n +
X
n
X
i,j=q,g
Oij
ns
≡ X
n,¯ n
X
i,j=q,g
OiB
n
1 P2
⊥
OBC
s
1 P2
⊥
OjC
¯ n +
X
n
X
i,j=q,g
OiB
n
1 P2
⊥
OjnB
s
.
Final Glauber action Three rapidity sectors Two rapidity sectors
The form of the collinear operators are fixed but the soft can have a much more general form Need to match up to 2 gluons to fix all
O1 = Pµ
?ST n S¯ nP?µ
O2 = Pµ
?ST ¯ n SnP?µ
O3 = P?·(g e Bn
S?)(ST n S¯ n) + (ST n S¯ n)(g e
B¯
n S?)·P?
O4 = P?·(g e B¯
n S?)(ST ¯ n Sn) + (ST ¯ n Sn)(g e
Bn
S?)·P?
O5 = P?
µ (ST n S¯ n)(g e
B¯
nµ S?) + (g e
Bnµ
S?)(ST n S¯ n)P? µ
O6 = P?
µ (ST ¯ n Sn)(g e
Bnµ
S?) + (g e
B¯
nµ S?)(ST ¯ n Sn)P? µ
O7 = (gBnµ
S?)ST n S¯ n(gB¯ n S?µ)
O8 = (gB¯
nµ S?)ST ¯ n Sn(gBn S?µ)
O9 = ST
n nµ¯
nν(ig e Gµν
s )S¯ n
O10 = ST
¯ n nµ¯
nν(ig e Gµν
s )Sn
OAB
s
= 8παs X
i
CiOAB
i
37
a)
p
μ,A
q
2n
p3n
G
qG k
p1n p4n
n n
p
μ,A
q
2n
p3n
G
k
p1n p4n
n n
p
1n
p
μ,A
q
2n
p3n
G
k
+
k p1n p4n
n n
p
4n
b)
n n n n n n
μ,A
n n n n n
μ,A
Not at all obvious that one collinear emission can be matched given that there are non-local TOP’s which contribute in the EFT non-locality only eliminated after using the on-shell condition
k · A = 0
similar matching works for gluon operators from Wilson line
39
a)
n n n n s q q '
n n n n
s q
n n n n
s q
n n n n
s q '
n n n n
s q '
b)
n n n n
S
μ,C
=
n n n n
S
μ,C
q q '
Matching all polarizations w/o using on shell conditions at 1-gluon (simplifies 2 gluon matching)
First row is reproduced by TOP’s in EFT
a)
n n n n
s s n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
s
n n n n
s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
s
n n n n
s s
n n n n
s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
n n n n
s s
b)
n n n n
s s
n n n n
s s s
n n n n
s s s
n n n n
s s
TOP’s
OBC
s
= 8⇡↵s ⇢ Pµ
⊥ST n S¯ nP⊥µ − P⊥ µ g e
Bnµ
S⊥ST n S¯ n − ST n S¯ ng e
B¯
nµ S⊥P⊥ µ − g e
Bnµ
S⊥ST n S¯ ng e
B¯
n S⊥µ
− nµ¯ nν 2 ST
n ig e
Gµν
s S¯ n
BC .
At one gluon level this operator reproduces the Lipatov vertex and generalizes it to arbitrary number of gluons. The form is uniquely fixed to all loops as there are no hard corrections to the theory.
C2 = C4 = C5 = C6 = C8 = C10 = 0 , C1 = −C3 = −C7 = +1 , C9 = −1 2 .
26
n n n n
q k+ k p k+ p k-
x x y y
n n n n
q k+ k p k+ p k-
x y y
Glauber Loop Exegesis: Two insertions of
IGbox = Z d
− d−2k⊥ d −k+ d −k−
2(~ k 2
⊥)(~
k⊥+~ q⊥)2 ⇣ k++p+
3 −(~
k⊥+~ q⊥/2) 2/p−
2 +i0
⌘⇣ −k−+p−
4 −(~
k⊥+~ q⊥/2) 2/p+
1 +i0
⌘ , IGcbox = Z d
− d−2k⊥ d −k+ d −k−
2(~ k 2
⊥)(~
k⊥+~ q⊥)2 ⇣ k++p+
3 −(~
k⊥+~ q⊥/2) 2/p−
2 +i0
⌘⇣ +k−+p−
1 −(~
k⊥+~ q⊥/2) 2/p+
1 +i0
⌘ . (46)
Ons¯
n
Neither integral is well defined. In the Abelian limit the sum of the two integrals is well defined. However to calculate the piece we need the individual diagrams to be well defined
CA
Introduce a rapidity regulator | 2q3/ν |−η
IGcbox = Z d
− d−2k⊥ d −k0 d −kz |kz|−2⌘ (⌫/2)2⌘
(~ k 2
⊥)(~
k⊥+~ q⊥)2 ⇣ k0− kz+p+
3 −(~
k⊥+ ~
q⊥ 2 )2/p− 2 +i0
⌘⇣ k0+ kz+p−
1 −(~
k⊥+ ~
q⊥ 2 )2/p+ 1 +i0
⌘ IGbox = Z d
− d−2k⊥ d −k0 d −kz |kz|−2⌘ (⌫/2)2⌘
(~ k 2
⊥)(~
k⊥+~ q⊥)2 ⇣ k0−kz+p+
3 −(~
k⊥+ ~
q⊥ 2 ) 2/p− 2 +i0
⌘⇣ −k0−kz+p−
4 −(~
k⊥+ ~
q⊥ 2 ) 2/p+ 1 +i0
⌘ Z
− d−2 − z | z|−2⌘ 2⌘
Z
⊥ ⊥ ⊥
h = ⇣−i 4⇡ ⌘ Z d
− d−2k⊥
(~ k 2
⊥)(~
k⊥+~ q⊥)2 h − i⇡ + O(⌘) i .
First term in build up of Glauber Phase Is there a Non-Abelian contribution to the phase? Yes coming from soft loops The Abelian phase is universal as it does not care about the the spin of the collinear lines.
We can also consider soft-collinear forward scattering
n n
S
q k+ k p k+ p k-
S
n n
q k+ k p k+ p k-
S
Just the boost of the previous case, yields the same result but now the the glaubers carry
(λ2, λ, λ2)
| k+ − βk− |η
Can tweak regulator to insure homogenous scaling
Gives same result as collinear-collinear
Other source of rapidity divergences are the Wilson lines which need to be regulated
Sn = X
perms
exp ⇢ −g n · P w|2Pz|−η/2 ⌫−η/2 n · As
S¯
n =
X
perms
exp ⇢ −g ¯ n · P w|2Pz|−η/2 ⌫−η/2 ¯ n · As
(56) Wn = X
perms
exp ⇢ −g ¯ n · P w2|¯ n · P|−η ⌫−η ¯ n · An
W¯
n =
X
perms
exp ⇢ −g n · P w2|n · P|−η ⌫−η n · A¯
n
Note we regulate every gluon to be consistent with the Glaubers (important for zero bin cancellation)
Soft, Collinear and Glauber all overlap
, Cn = Cn C(S)
n
C(G)
n
+ C(GS)
n
Using the rapidity regulator many of the zero- bin diagrams vanish but crucially not all as in this example.
S
G
S(G) = G This is why we don’t see the Glauber in hard matching
49
p p p p
n n n n
n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
Full Theory Graphs
s
t Sn¯
n 1
8i⇡ ln ⇣ −t m2 ⌘ + i↵2
s
t Sn¯
n 2
− 4 ln2 ⇣m2 −t ⌘ − 12 ln ⇣m2 −t ⌘ − 14
s
t Sn¯
n 3
− 4 ln ⇣ s −t ⌘ ln ⇣ −t m2 ⌘ + 22 3 ln ⇣ µ2 −t ⌘ + 170 9 + 2⇡2 3
s
t Sn¯
n 4
− 8 3 ln ⇣ µ2 −t ⌘ − 40 9
(
Sn¯
n 1
= h ¯ unT AT B ¯ n / 2 un ih ¯ v¯
nT BT A n
/ 2 v¯
n
i , Sn¯
n 2
= CF h ¯ unT A ¯ n / 2 un ih ¯ v¯
nT A n
/ 2 v¯
n
i , Sn¯
n 3
= CA h ¯ unT A ¯ n / 2 un ih ¯ v¯
nT A n
/ 2 v¯
n
i , Sn¯
n 4
= TF nf h ¯ unT A ¯ n / 2 un ih ¯ v¯
nT A n
/ 2 v¯
n
i .
51
n n n n
G G
n n n n
G G
n n n n
S
n n n n
S
n n n n
S
n n n n n n n n n n n n
n n n n n n n n n n
n n n
n n n n n n n n n n n n
n n n n n n n n n n
n n n
n n n n n n n n n n n n n n n n n n n n
EFT Diagrams
n n n n n n n
n n n n n n n
n n n n n n n
n n n n n n n
Killed by soft zero-bin subtractions
= i↵2
s
t Sn¯
n 3
⇢ − 4 ⌘h(✏, µ2/m2) − 4 ⌘ g(✏, µ2/t) − 4 ln ⇣ µ2 4⌫2 ⌘ ln ⇣m2 −t ⌘ − 2 ln2 ⇣ µ2 m2 ⌘ + 2 ln2 ⇣ µ2 −t ⌘ − 2⇡2 3 + 22 3 ln µ2 −t + 134 9
s
t Sn¯
n 4
− 8 3 ln ⇣ µ2 −t ⌘ − 40 9
(108)
Soft Total Includes counter term for α Note: no UV poles Collinear total
ln s/ν2 = ln(n · p/ν) + ln(¯ n · p/ν)
combining collinear sectors
Total EFT
c.t. = i↵2
s
t Sn¯
n 1
8i⇡ ln ⇣ t m2 ⌘ + i↵2
s
t Sn¯
n 2
4 ln2 ⇣m2 t ⌘ 12 ln ⇣m2 t ⌘ 14
s
t Sn¯
n 3
⇢ 4 ln ⇣ s t ⌘ ln ⇣ t m2 ⌘ + 22 3 ln µ2 t + 170 9 + 2⇡2 3
s
t Sn¯
n 4
8 3 ln ⇣ µ2 t ⌘ 40 9
(113) = i↵2
s
t Sn¯
n 3
⇢4 ⌘h(✏, µ2/m2) + 4 ⌘ g(✏, µ2/t) + 4 ln ⇣4⌫2 s ⌘ ln ⇣ t m2 ⌘ + 2 ln2 ⇣m2 t ⌘ +4+ 4⇡2 3
s
t Sn¯
n 2
4 ln2 ⇣m2 t ⌘ 12 ln ⇣m2 t ⌘ 14
(112)
running of Glauber potential
=full, no hard matching!
Even though the rapidity divergences cancel we must renormalize the factorized operators distinctly
→ ~ OBbare
¯ n
= ˆ VO¯
n · ~
OB
¯ n (⌫, µ) ,
ˆ VO¯
n =
B @ 1 + V qq
¯ n
V qg
¯ n
V gq
¯ n
1 + V gg
¯ n
1 C A , ~ OB
¯ n =
B @ OqB
¯ n
OgB
¯ n
1 C A .
Collinear mixing:
~ OAbare
sn
= ˆ VOsn · ~ OA
sn(⌫, µ) ,
ˆ VOsn = B @ 1 + V qq
sn
V qg
sn
V gq
sn
1 + V gg
sn
1 C A , ~ OA
sn =
B @ OqnA
s
OgnA
s
1 C A ,
Soft single index ops:
~ OABbare
s
= ˆ VOs · ~ OAB
s
(⌫, µ) , ˆ VOs = B B B B B B B B B B B B @ 1 + Vs V Tqq
s
ˆ VOsn ⊗ ˆ VOs¯
n
V Tgq
s
V Tgg
s
V Tqg
s
1 C C C C C C C C C C C C A , ~ OAB
s
= B B B B B B B B B B B B @ OAB
s
i R d4x T OqnA
s
(x)Oq¯
nB
s
(0) i R d4x T OgnA
s
(x)Oq¯
nB
s
(0) i R d4x T OgnA
s
(x)Og¯
nB
s
(0) i R d4x T OgnA
s
(x)Oq¯
nB
s
(0) 1 C C C C C C C C C C C C A .
Soft double index ops:
Re-sum the logs in parton-parton scattering, simplest to run collinear sectors (no TOPs)
ν d dν (OqA
n + OgA n ) = γnν(OqA n + OgA n ) ,
ν d dν (OqnA
s
+ OgnA
s
) = γsν(OqnA
s
+ OgnA
s
) .
Structure is Fixed So all we need is γν
n
Which we can extract from matching calculation M(ν = √s) ∼ (s/t)γ Gluon Reggeization
γnν ≡ γqq
nν + γgq nν = γgg nν + γqg nν ,
mixing leads to universality
γ = αsCA 2π Log(−t/m2)
Summing Logs in IR safe quantities
Consider observables for which the soft radiation is not
BKFL equation
Consider the total cross section for hadron scattering try to capture s dependence pp = Z d dt dt ⇠ Z dt X
Xs,Xn,X¯
n
hpn¯ pn | Xs, Xn, X¯
niO(t(Pn ?, P ¯ n ?))hXs, Xn, X¯ n | pn¯
pni,
Running the soft function of Glauber operator insert LII
G
GABA0B0(q?, q0
?) =
X
X
⌦
s(1,1)(q?, q0 ?)
↵⌦ X
s(1,1)(q?, q0 ?)
OAB
s(1,1)(q?, q0 ?) = OAB s
(q?, q0
?) +
Z d4x T ⇥ OqnA
s
(q?) + OgnA
s
(q?) ⇤⇥ Oq¯
nA
s
(q0
?) + Og¯
nA
s
(q0
?)
⇤ .
q q ' = G(0)abcd(l?, l0
?) ⌘ Oab s | 0ih0 | Ocd s =
✓8⇡↵ l2
?
◆2 abcd2(l? l0
?) ,
S
q q '
+
S
q q '
= ↵ 2⇡2 aCAΓ[⌘] Z d2q? G(0)(l?, l0
?)l2 ?
q2
?(q? l?)2
(
˜ G(l?, l0
?) = l2 ?G(l?, l0 ?)l0 ? 2
Here used:
q q ' S S
= (2)8α3π2fabefcdeΓ[η] Z [d2q?] (q? l?)2l2
?q2 ?
δ2(q? l 0
?)
= α π2 CAΓ[η] Z [d2q?]q2
?
(q? l?)2l2
?
G(0)abcd(q?, l 0
?) ,
˜ Gren(~ l?,~ l0
?) =
Z d2q?Z(l?, q?) ˜ Gbare(q?, l0
?)
ν d dν Gren(q?, l0
?) =
Z d2q0
?γ(q?, q0 ?)Gren(q0 ?, l0 ?)
dimension is given by
γ(q?, q0
?) =
Z d2k?Z(q?, k?)ν d dν Z1(k?, l0
?) = αsCA
π2 ✓ 1 (q? − q0
?) − δ2(q? − q0 ?)
Z d2l? q2
?
2l2
?(l? − q?)2
◆ (187)
BFKL equation ˜ ˜ Soft functions evolution is given by the BFKL equation
Open Directions: Indenumerable