Assorted NLL small-x comments (with emphasis on ... · Assorted NLL small-x comments (with emphasis on preasymptotics) Gavin Salam LPTHE — Univ. Paris VI & VII and CNRS In collaboration

Assorted NLL small- xxx comments(with emphasis on preasymptotics)

Gavin Salam

LPTHE — Univ. Paris VI & VII and CNRS

In collaboration with M. Ciafaloni, D. Colferai and A. Stasto

QCD at cosmic energiesErice, August 30 – September 4, 2004

Assorted NLL small-xxx comments (with emphasis on preasymptotics) – p.1/16

Introduction

Contents inspired by discussion during workshop

Relative importance of running coupling versus higher orders (cf. Mueller)Many features common to all problems with cutoffs:

saturationsplitting function (will explain why ≡ cutoff)


Introduction




Give discussion in context of CCSS approach (cf. Ciafaloni)

NLL BFKL supplemented with DGLAP effectsnumerical solutions of resulting equations (‘no approximations’)extraction of splitting function


Introduction




Give discussion in context of CCSS approach (cf. Ciafaloni)

NLL BFKL supplemented with DGLAP effectsnumerical solutions of resulting equations (‘no approximations’)extraction of splitting function

Characteristic result: significant preasymptotic effectsimpact on phenomenology? (question by Strikman)

convolution of splitting function with CTEQ gluon


Improved NLL x? Start with kernel. . .

αs + α2s

x lnx0

x

x0

x � x0

αs x lnQ2

Q20

Q2

0

Q2 � Q2

0

+ α2s

BFKL

DGLAP

anti-DGLAP+ Q2 ⇔ Q2

0



αs + α2s

x lnx0

x

x0

x � x0

ln x ln x

αs x lnQ2

Q20

Q2

0

Q2 � Q2

0

+ α2s

ln Q2 ln2 Q2

BFKL

DGLAP

anti-DGLAP+ Q2 ⇔ Q2

0



αs x lnQ2

Q20

Q2

0

Q2 � Q2

0

+ α2s

combinedBFKL+DGLAP

kernel K

αs + α2s

x lnx0

x

x0

x � x0

ln x ln x

ln Q2 ln2 Q2

BFKL

DGLAP

anti-DGLAP

(bewaredouble counting)

+ Q2 ⇔ Q2

0

GPS, Ciafaloni, Colferai ’98–99

ibid. + Stasto ’03


Iteration of kernel ⇒ Green function

x0, Q2

0

x,Q2

Green function: G(

ln xx0

;Q0, Q)


Iteration of kernel ⇒ Green function

x0, Q2

0

x,Q2

Green function: G(

ln xx0

;Q0, Q)

0.1

1

10

0 5 10 15 20

2π k

02 G(Y

; k0

+ ε,

k0)

Y

k0 = 20 GeV

(a)

LLscheme Ascheme B


Green function ⇒ effective DGLAP splitting function

Construct a gluon density from Green function (take k � k0):

xg(x,Q2) ≡∫ Q

d2k G(ν0=k2)(ln 1/x, k, k0)




xg(x,Q2) ≡∫ Q

d2k G(ν0=k2)(ln 1/x, k, k0)

Numerically solve equation for effective splitting function, Pgg,eff(z,Q2) :

dg(x,Q2)

d ln Q2=

∫dz

zPgg,eff(z,Q2) g

(x

z,Q2

)




xg(x,Q2) ≡∫ Q

d2k G(ν0=k2)(ln 1/x, k, k0)


dg(x,Q2)

d ln Q2=

∫dz

zPgg,eff(z,Q2) g

(x

z,Q2

)

Factorisation

Splitting function:red paths

Green function:all paths

x

Q1

Q2

g(x,Q )1

g(x,Q2 )

k

2

2

Pgg

Evolution paths in x,k

factorized (non−perturbative)

g




xg(x,Q2) ≡∫ Q

d2k G(ν0=k2)(ln 1/x, k, k0)


dg(x,Q2)

d ln Q2=

∫dz

zPgg,eff(z,Q2) g

(x

z,Q2

)

Factorisation

Splitting function:red paths

Green function:all paths

Splitting function ≡evolution with cutoff x

Q1

Q2

g(x,Q )1

g(x,Q2 )

k

2

2

Pgg

Evolution paths in x,k

factorized (non−perturbative)

g


BFKL ‘power’

Two classes of correction, to power growth ω:

ω = 4 ln 2 αs(Q2)

1 − 6.5 αs︸︷︷︸

NLL

− 4.0 α2/3s

︸︷︷︸

running

+ · · ·

αs = αsNc/π

NLL piece is universal

running piece appears only in problems with cutoffs

a consequence of asymmetrydue to cutoff (only scales higherthan cutoff contribute)

αs(Q2) → αs(Q

2e−X/(bαs)1/3

)

Hancock & Ross ’92


BFKL ‘power’

Two classes of correction, to power growth ω:

ω = 4 ln 2 αs(Q2)

1 − 6.5 αs︸︷︷︸

NLL

− 4.0 α2/3s

︸︷︷︸

running

+ · · ·

αs = αsNc/π

NLL piece is universal

running piece appears only in problems with cutoffs

a consequence of asymmetrydue to cutoff (only scales higherthan cutoff contribute)

αs(Q2) → αs(Q

2e−X/(bαs)1/3

)

Hancock & Ross ’92

Beyond first terms, not possible to separate effects of ‘pure’ higher orders &running coupling


BFKL splitting function power growth – results

ωc

(spl

ittin

g fn

)

αs

Q [GeV]

LL, fixed αs

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3

1.93.26.824300



ωc

(spl

ittin

g fn

)

αs

Q [GeV]

LL, fixed αsLL, running αs

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3

1.93.26.824300



ωc

(spl

ittin

g fn

)

αs

Q [GeV]

LL, fixed αsLL, running αsNLLB, fixed αs

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3

1.93.26.824300



ωc

(spl

ittin

g fn

)

αs

Q [GeV]

LL, fixed αsLL, running αsNLLB, fixed αs

NLLB, running αs 0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3

1.93.26.824300


Can one neglect NLL terms? Examine full Pgg(z) splitting fn

0.1

1

10-10 10-8 10-6 10-4 10-2 100

z P

(z)

z

Q = 4.5 GeVα−s(Q

2) = 0.215

LL (fixed α−s)

LO DGLAP



0.1

1

10-10 10-8 10-6 10-4 10-2 100

z P

(z)

z

Q = 4.5 GeVα−s(Q

2) = 0.215

LL (fixed α−s)

LL (α−s(q2))

LO DGLAP



0.1

1

10-10 10-8 10-6 10-4 10-2 100

z P

(z)

z

Q = 4.5 GeVα−s(Q

2) = 0.215

LL (fixed α−s)

LL (α−s(q2))

NLLB

LO DGLAP


Interim conclusion

Individually, running coupling and NLL effects are large

BFKL ‘power’ has only moderate extra suppression when combining bothnon-linearities between higher-orders and running coupling

αs(Q2) → αs(Q

2e−X/(bαs)1/3

) − Cα2s(Q2e−X/(bαs)1/3

)


Interim conclusion



αs(Q2) → αs(Q

2e−X/(bαs)1/3

) − Cα2s(Q2e−X/(bαs)1/3

)

BUT: power is not the most relevant characteristic for forseeable energies

good estimate of splitting function requires all effects (running, NLL) to beaccounted for


Interim conclusion



αs(Q2) → αs(Q

2e−X/(bαs)1/3

) − Cα2s(Q2e−X/(bαs)1/3

)

BUT: power is not the most relevant characteristic for forseeable energies

good estimate of splitting function requires all effects (running, NLL) to beaccounted for

Likely to be true also for saturation scale Q2s(x). . .


Dominant phenomenological structure is dip

Rapid rise in Pgg is not for today’senergies!

Main feature is a dip at x ∼ 10−3

z P

gg(z

)

z

ω-expansion (1999)NLLB (2003)

LO DGLAP

0

0.1

0.2

0.3

0.4

0.5

10-6 10-5 10-4 10-3 10-2 10-1 100

Q = 4.5 GeVα−s(Q

2) = 0.215

1/4 < µ2/Q2 < 4





Questions:

Various ‘dips’ have been seenThorne ’99, ’01 (running αs, NLLx)

ABF ’99–’03 (fits, running αs)CCSS ’01,’03 (running αs, NLLB)

Is it always the same dip?

z P

gg(z

)

z


LO DGLAP

0

0.1

0.2

0.3

0.4

0.5

10-6 10-5 10-4 10-3 10-2 10-1 100

Q = 4.5 GeVα−s(Q

2) = 0.215

1/4 < µ2/Q2 < 4





Questions:




Is the dip a rigorous prediction?z P

gg(z

)

z


LO DGLAP

0

0.1

0.2

0.3

0.4

0.5

10-6 10-5 10-4 10-3 10-2 10-1 100

Q = 4.5 GeVα−s(Q

2) = 0.215

1/4 < µ2/Q2 < 4





Questions:




Is the dip a rigorous prediction?

What is its origin?Running αs, momentum sum rule. . . ?

z P

gg(z

)

z


LO DGLAP

0

0.1

0.2

0.3

0.4

0.5

10-6 10-5 10-4 10-3 10-2 10-1 100

Q = 4.5 GeVα−s(Q

2) = 0.215

1/4 < µ2/Q2 < 4





Questions:




Is the dip a rigorous prediction?

What is its origin?Running αs, momentum sum rule. . . ?

NNLO DGLAP gives a clue. . .

−1.54 α3sln 1

x

z P

gg(z

)

z


LO DGLAP

NNLO DGLAP

0

0.1

0.2

0.3

0.4

0.5

10-6 10-5 10-4 10-3 10-2 10-1 100

Q = 4.5 GeVα−s(Q

2) = 0.215

1/4 < µ2/Q2 < 4


Reorganise perturbative series

ln 1/x

ln 2 1/x

ln 3 1/x

αs

αs

αs

αs

αs

const.

LLx NLLx NNLLx . . .

0

0

x

0

−−

−

x

nf

x

x

x

x

x

x

. . .

2

3

4

5



ln 1/x

ln 2 1/x

ln 3 1/x

αs

αs

αs

αs

αs

const.


0

0

x

0

−−

−

x

x

x

x

x

. . .

nf

x

x

2

3

4

5

10.110-210-310-4

x Pgg

(x)

x

α−s = 0.05



ln 1/x

ln 2 1/x

ln 3 1/x

αs

αs

αs

αs

αs

const.


0

0

x

0

−−

−

x

x

x

x

. . .

x

x

nf

x

2

3

4

5

At moderately small x, first terms withx-dependence are

−1.54 α3

sln

1

x

10.110-210-310-4

x Pgg

(x)

x

α−s = 0.05



ln 1/x

ln 2 1/x

ln 3 1/x

αs

αs

αs

αs

αs

const.


0

0

0

−−

−

x

x

x

x

. . .

x

x

x

nf

x

2

3

4

5


−1.54 α3sln

1

x+ 0.401 α4

sln3

1

x

10.110-210-310-4

x Pgg

(x)

x

α−s = 0.05



ln 1/x

ln 2 1/x

ln 3 1/x

αs

αs

αs

αs

αs

const.


0

0

0

−−

−

x

x

x

x

. . .

x

x

x

nf

x

2

3

4

5


−1.54 α3sln

1

x+ 0.401 α4

sln3 1

x

Minimum when

αs ln2 x ∼ 1 ≡ ln1

x∼ 1

√αs

10.110-210-310-4

x Pgg

(x)

x

α−s = 0.05

∼ α− s5/2


Systematic expansion in√

αs

ln 1/x

ln 2 1/x

ln 3 1/x

αs

αs

αs

αs

αs

const.


0

0

0

−−

−

x

x

x

x

. . .

x

x

x

nf

x

2

3

4

5

Position of dip

ln1

xmin' 1.156√

αs

Depth of dip

−d ' −1.237α5/2s



αs

ln 1/x

ln 2 1/x

ln 3 1/x

αs

αs

αs

αs

αs

const.


0

0

−−

−

x

x

x

. . .

x

x

x

nf

x

0

x

2

3

4

5

Position of dip

ln1

xmin' 1.156√

αs

+ 6.947

Depth of dip

−d ' −1.237α5/2s

− 11.15α3s



αs

ln 1/x

ln 2 1/x

ln 3 1/x

αs

αs

αs

αs

αs

const.


0

0

−−

−

x

x

x

. . .

x

x

x

nf

x

0

x

2

3

4

5

Position of dip

ln1

xmin' 1.156√

αs

+ 6.947 + · · ·

Depth of dip

−d ' −1.237α5/2s

− 11.15α3s+ · · ·

NB:

convergence is very poorAs ever at small x!

higher-order terms in expansionneed NNLLx info


Phenomenological impact?

Phenomenological relevance comes through impact on growth of small-x gluonwith Q2.

∂g(x,Q2)

d ln Q2= Pgg ⊗ g + Pgq ⊗ q


Phenomenological impact?

Phenomenological relevance comes through impact on growth of small-x gluonwith Q2.

∂g(x,Q2)

d ln Q2= Pgg ⊗ g + Pgq ⊗ q

At small x, Pgg ⊗ g dominates.

Take CTEQ6M gluon as ‘test’ case for convolution.

Because it’s nicely behaved at small-x


Phenomenological impact? Pgg ⊗ g(x)z P

gg(z

)

z

NLLB (2003)

LO DGLAP

NNLO DGLAP

0

0.1

0.2

0.3

0.4

0.5

10-6 10-5 10-4 10-3 10-2 10-1 100

Q = 2.0 GeV

α−s(Q2) = 0.30

⊗

0

2

4

6

8

10

12

10-6 10-5 10-4 10-3 10-2 10-1 100

g(x

,Q2)

x

gluon = CTEQ6M

Q = 2.0 GeV

CTEQ 6M gluon


Phenomenological impact? Pgg ⊗ g(x)

z P

gg

(z)

zNLLB (2003)

LO DGLAP

NNLO DGLAP

0 0.1

0.2 0.3

0.4 0.5

10-610-510-410-310-210-1100

Q = 2.0 GeV

α−s(Q2) = 0.30

×

0

2

4

6

8

10

12

10-6 10-5 10-4 10-3 10-2 10-1 100

g(x

,Q2)

x

gluon = CTEQ6M

Q = 2.0 GeV

CTEQ 6M gluon


Pgg ⊗ g(x)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

(Pgg

x g

) / g

x

gluon = CTEQ6M

Q = 2.0 GeV

LO DGLAP

NLO DGLAP

NNLO DGLAP


Pgg ⊗ g(x)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

(Pgg

x g

) / g

x

gluon = CTEQ6M

Q = 2.0 GeV

LO DGLAP

NLO DGLAP

NNLO DGLAP

small-x rsmd


Conclusions

Quantities such as ‘BFKL power’ are nice for discussing certain asymptoticproperties

BUT: Preasymptotic effects cannot be neglected, often even for cosmic-rayenergies.

Specifically for Pgg

Pgg has dip, strongly influenced by NNLO DGLAP

For low Q, after convolution with gluon, dip and rise compensate →similar to NLO!


Documents

Assorted NLL small-x comments (with emphasis on ... · Assorted NLL small-x comments (with emphasis on preasymptotics) Gavin Salam LPTHE — Univ. Paris VI & VII and CNRS In collaboration