Resummation of large transverse - LPTHE · Resummation of large transverse logarithms D. Triantafyllopoulos, ECT*/FBK 9 Resummation of double logs (DLA) Resum double logs to all orders

Resummation of large transverse logarithms in the high energy evolution of color dipoles

QCD at Cosmic Energies VIIChalkida , Greece

16-20 May 2016

Dionysios TriantafyllopoulosECT*/FBK, Trento, Italy

E. Iancu, J.D. Madrigal, A.H. Mueller, G. Soyez and D.T., Phys. Lett. B744 (2015) 293, B750 (2015) 643 and w.i.p.

Outline

Resummation of large transverse logarithms D. Triantafyllopoulos, ECT*/FBK

❖ NLO BK equation and large transverse logarithms

❖ Unphysical solutions

❖ Resummation of logarithms to all orders

❖ Restoration of stability, numerical solutions

❖ Fits to HERA data

2

CGC and BK in Heavy Ion Collisions


❖ CGC: high energy evolution of hadronic wave function

❖ Best d.o.f. : Wilson lines for projectile partons scattering

❖ Cross section in DIS, single/double particle production at forward rapidity in pA collisions, energy density just after an AA collision, … : Local in rapidity observables: correlators of Wilson lines.

❖ To excellent accuracy, all such correlators expressed in terms of dipole scattering. BK equation.

3


Diagrams for dipole evolution

4

❖ LO

❖ NLO Nf

❖ NLO Nc

❖ First emission soft, second non-soft in general

The BK equation at NLO

Resummation of large transverse logarithms D. Triantafyllopoulos, ECT*/FBK 5

b̄ =11

12� 1

6

Nf

NcV †i = Pexp

ig

Zdz+A�

a (z+, zi)t

a

�Sij =

1

Nctr�V †i Vj

�

dS12

dY=

↵̄s

2⇡

Zd2z3

z212z213z

223

1 + ↵̄s

✓b̄ ln z212µ

2 � b̄z213 � z223

z212ln

z213z223

+67

36� ⇡2

12� 5

18

Nf

Nc

� 1

2ln

z213z212

lnz223z212

◆�(S13S32 � S12)

+↵̄2s

8⇡2

Zd2z3 d2z4

z434

�2 +

z213z224 + z214z

223 � 4z212z

234

z213z224 � z214z

223

lnz213z

224

z214z223

+z212z

234

z213z224

✓1 +

z212z234

z213z224 � z214z

223

◆ln

z213z224

z214z223

�

S13S34S42�

1

2N3c

tr�V1V

†3 V4V

†2 V3V

†4

�� 1

2N3c

tr�V1V

†4 V3V

†2 V4V

†3

�� S13S32+

1

N2c

S12

�

+↵̄2s

8⇡2

Nf

Nc

Zd2z3 d2z4

z434

2� z213z

224 + z214z

223 � z212z

234

z213z224 � z214z

223

lnz213z

224

z214z223

�

S14S32�

1

N3c

tr�V1V

†2 V3V

†4

�� 1

N3c

tr�V1V

†4 V3V

†2

�+

1

N2c

S12S34 � S13S32+1

N2c

S12

�

zij = zi�zj

Large transverse logs


❖ Strongly ordered large ``perturbative’’ dipoles (DLA)

❖ Simplest thing: one step for GBW-type initial condition

❖ Large dipoles interact stronger, real terms dominate

❖ NLO correction > LO term, unstable expansion in αs

❖ Expect unstable numerical solution, as in BFKL

dT12

dY= ↵̄s

Z 1/Q2s

z212

dz213z212z413

✓1� ↵̄s

1

2ln2

z213z212

� ↵̄s11

12ln

z213z212

◆T13

T12 =

(z212Q

2s0 for z12Qs0 ⌧ 1

1 for z12Qs0 � 1

) �T12 ' �Y ↵̄sz212Q

2s0

✓ln

1

z212Q2s0

� ↵̄s1

6

ln

3 1

z212Q2s0

� ↵̄s11

24

ln

2 1

z212Q2s0

◆

1/Qs � z14 ' z24 ' z34 � z13 ' z23 & z12 exp�↵̄�1/2,�1s

�� z12

Unstable numerical solutions


0.0001

0.001

0.01

0.1

1

10-2 10-1 100 101 102 103 104 105

F(Y,

k)

k2 (GeV2)

10−5 10−4 10−3 10−2 10−1 100

rΛQCD

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

(∂yN)/N

Qs,0/ΛQCD = 4Qs,0/ΛQCD = 13

Qs,0/ΛQCD = 19Qs,0/ΛQCD = 26

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

-5 0 5 10 15 20 25

T(L

,Y)

L=log(1/r2)

(a) LO

α-

s=0.25

Y=0Y=4Y=8Y=12Y=16

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

-5 0 5 10 15 20 25

T(L

,Y)

L=log(1/r2)

(b) NLO

α-

s=0.25

Y=0Y=4Y=8


Two gluons and time ordering (kinematics)

❖ Take k ⌧ p and k+ ⌧ p+

�T12 = �g4N2c

(2⇡2)

Zdp+

p+

Zdk+

k+

Zd2z3d

2z4

Z

pkp̃k̃

p · p̃ k · k̃p2p̃2k2k̃2

eip·z31+ip̃·z13+ik·z42+ik̃·z34

h1 + k+

p+p2

k2

ih1 + k+

p+(p̃�k̃)2

k̃2

i 2T (z4)

❖ Largest logs when time ordering ⌧k = k+/k2 ⌧ ⌧p = p+/p2

❖ Add all diagrams: WW kernels, phases and trans. mom. int. → two dipole kernels. For z4 � z3 � z12

�T12 = ↵̄2s

Zdp+

p+dk+

k+⇥⇣p+

z23z24

� k+⌘dz23dz

24

z212z23z

44

T (z4) ! � ↵̄2s�Y

2

Z

z212

dz24z44

ln2z24z212

T (z4)


Resummation of double logs (DLA)❖ Resum double logs to all orders in non-local equation

❖ I.C. →

❖ Leads to physical amplitude T for Y>ρ

T (0, ⇢) = e�⇢

T (Y, ⇢) = T (0, ⇢) + ↵̄s

Z ⇢

0d⇢1

Z Y�⇢+⇢1

0dY1 e

�(⇢�⇢1)T (0, ⇢1), ⇢ = ln 1/z212Q20

❖ Mathematically equivalent to local equation

T̃ (Y, ⇢) = T̃ (0, ⇢) + ↵̄s

Z Y

0dY1

Z ⇢

0d⇢1 e

�(⇢�⇢1)J1�2p↵̄s(⇢� ⇢1)2

�p

↵̄s(⇢� ⇢1)2T̃ (0, ⇢1)

T̃ (0, ⇢) = e�⇢J1�2p

↵̄s⇢2�


Double logs resummed in BK

❖ Agrees with NLO BK (double log term) when truncated to order ↵̄2

s

❖ Resums double log terms to all orders

❖ Solution should behave nicely

❖ Initial condition: T̃ (0, ⇢) = 1� e�T̃2g(0,⇢)

dT̃12

dY=

↵̄s

2⇡

Zd2z3

z212z213z

223

J1�2p↵̄sL13L23

�p↵̄sL13L23

(T̃13 + T̃23 � T̃12 � T̃13T̃23)


Numerical solution

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

α-s

χ(γ

)

γ

symbols: numerics

lines: analytic

α-

s=0.25

LONLONLO+resum 10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

-5 0 5 10 15 20 25

T(L

,Y)

L=log(1/r2)

(c) NLO+resum

α-

s=0.25

Y=0Y=4Y=8Y=12Y=16

BFKL on z2�12 ⌘ r2�

!LO =↵̄s

�+

↵̄s

1� �+ finite !NLO =

↵̄s

�+

↵̄s

1� �� ↵̄2

s

(1� �)3+ finite

!resNLO = !NLO � ↵̄s

1� �+

↵̄2s

(1� �)3+

1

2

h�(1� �) +

p(1� �)2 + 4↵̄s

i+ finite


Numerical solution

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12 14 16 18 20

log[Q

2 s(Y

)/Q

2 s(0)]

Y

α-

s=0.25

LO

NLO+resum

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 2 4 6 8 10 12 14 16 18 20

dlo

g[Q

2 s(Y

)]/d

YY

α-

s=0.25

LO

NLO+resum

❖ Considerable speed reduction, roughly factor of 1/2

Single log in quark contribution (dynamics)❖ Take k ⌧ p and ⇣ = k+/p+ ⌧ 1

⌃Aij = 8↵2sNf

Zdp+

p+

Z 1

0d⇣

Z

pkp̃k̃

p · k p̃ · k̃ + p · p̃ k · k̃ � p · k̃ k · p̃p2p̃2(⇣p2 + k2)(⇣p̃2 + k̃2)

�eip·z31+ik·z41 � eip·z32+ik·z42

� ⇣eip̃·z13+ik̃·z14 � eip̃·z23+ik̃·z24

⌘

❖ Integrate transverse momenta

⌃Aij =↵2sNf

2⇡4�Y

Z 1

0d⇣

z212z24

z43 + ⇣2z44(z23 + ⇣z24)

4' ↵2

sNf

3⇡4�Y

z212z23z

44

z23 scales like ⇣z24 , no ordering in time ❖ Insert color structure and Wilson lines

�T12

�Y=

↵̄2s

6

Nf

Nc

✓1� 1

N2c

◆z212

Z 1

z212

dz24z44

lnz24z212

T (z4)

13Resummation of large transverse logarithms D. Triantafyllopoulos, ECT*/FBK


❖ Pole residues related to coefficients in splitting functions

CF

Nc

Z 1

0dz z!Pqg(z) =

Nf

6Nc� Nf

6N3c

+O(!)

Z 1

0dz z!Pgg(z) =

1

!� 11

12� Nf

6Nc+O(!)

❖ Starting from DGLAP, write in Mellin space

❖ Integrate ω → BFKL

Relationship to splitting functions

14

T12 ⇡Z

d�

2⇡i

d!

2⇡i

e!Y��⇢

1� � + ! � ↵̄s P̃ (!)| {z }' 1

!�A1

Resum single logs


❖ Keep 1/ω-A1 → local in Y equation

!coll

(�) =1

2

h�(1� � + ↵̄sA1

) +p(1� � + ↵̄sA1

)2 + 4↵̄s

i

❖ Extend to BK as in double logs case

dT̃12

dY=

↵̄s

2⇡

Zd2z3

z212z213z

223

✓z212z2<

◆±A1↵̄s J1�2p↵̄sL13L23

�p↵̄sL13L23

(T̃13 + T̃23 � T̃12 � T̃13T̃23)

z< = min{z13, z23}❖ . - sign when , resums logsz< < z12

extra factor in kernel: anomalous dimension e�A1↵̄s(⇢�⇢1)

Running coupling


❖ Choose μ to cancel potentially large log in all regionsLarge daughter dipoles : Small daughter dipoles : In general : Hardest scale

µ ⇡ 1/z12µ ⇡ 1/min{z13, z23}

µ ⇡ 1/min{zij} X❖ Balitsky-prescription: ✓, albeit unphysical slow

dT12

dY=

↵̄s(µ)

2⇡

Zd2z3

z212z213z

223

1 + ↵̄s(µ)

✓b̄ ln z212µ

2 � b̄z213 � z223

z212ln

z213z223

◆�

(T13 + T23 � T12 � T13T23)

❖ Choose coefficient of to vanish: ✓, better choiceb̄

↵s =

1

↵s(z12)+

z213 � z223z212

↵s(z13)� ↵s(z23)

↵s(z13)↵s(z23)

��1

Couplings comparison


α��α ��α��

� � � � �

��

��

��

��

|�-�|

|�-�|=�ϕ=�

α��α ��α��

� � � � �

��

��

��

��

|�-�|

|�-�|=�ϕ=π /�

α��α ��α��

� π /� π /� �π /� π

��

��

��

��

ϕ

|�-�|=�|�-�|=��

Numerical solution (fixed)


0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

dlo

g[Q

2 s(Y

)]/d

Y

Y

speed, α- s=0.25

LO

DLA at NLO

DLA resum

DLA+SL resum

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

-5 0 5 10 15 20 25

T(ρ

,Y)

ρ=log(1/r2)

DLA+SL resum, α- s=0.25

Y=0Y=4Y=8Y=12Y=16

Numerical solution (prescirption:small)


10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

-5 0 5 10 15 20 25

T(ρ

,Y)

ρ=log(1/r2)

DLA+SL resum, β0=0.72, smallest

Y=0Y=4Y=8Y=12Y=16

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

dlo

g[Q

2 s(Y

)]/d

Y

Y

speed, β0=0.72, smallest

LO

DLA at NLO

DLA resum

DLA+SL resum

Fit


theory

/data

0.9

0.95

1

1.05

1.1 0.045,0.065 0.085 0.1 0.15 0.2

theory

/data

x

0.9

0.95

1

1.05

1.1

10-6 10-5 10-4

0.25

x10-6 10-5 10-4

0.35

x10-6 10-5 10-4

0.4

x10-6 10-5 10-4

0.5

x10-6 10-5 10-4

0.65

theory

/data

0.9

0.95

1

1.05

1.1 0.85 1.2 1.5 2 2.7

theory

/data

x

0.9

0.95

1

1.05

1.1

10-5 10-4 10-3

3.5

x10-5 10-4 10-3

4.5

x10-5 10-4 10-3

6.5

x10-5 10-4 10-3

8.5

x10-5 10-4 10-3

10

theory

/data

0.9

0.95

1

1.05

1.112 15 18 22 27

theory

/data

x

0.940.960.98

11.021.041.06

10-3 10-2

35

x10-3 10-2

45 χ2

1.131.221.101.13

λs

0.220.240.200.23

HERA datasmall, DL+SLfac, DL+SLsmall, DLfac, DL

10-1

100

101

10-6 10-5 10-4 10-3 10-2

Q2 o

r 4/r2 [G

eV2 ]

x

dataGBW,smallGBW,newRun,smallRun,new

Fit


❖ Including single logs: more physical parameters

❖ rcMV model: can be extrapolated to higher Q2

❖ Small and ``new’’ prescription: good fit B-prescription: ``doesn’t make it’’

❖ No anomalous dimension in initial condition

init RC sing. �

2 per data point parameterscdt. schm logs �

red

�

cc̄red

FL Rp[fm] Q

0

[GeV] C↵ p C

MV

GBW small yes 1.135 0.552 0.596 0.699 0.428 2.358 2.802 -GBW fac yes 1.262 0.626 0.602 0.671 0.460 0.479 1.148 -rcMV small yes 1.126 0.578 0.592 0.711 0.530 2.714 0.456 0.896rcMV fac yes 1.222 0.658 0.595 0.681 0.566 0.517 0.535 1.550GBW small no 1.121 0.597 0.597 0.716 0.414 6.428 4.000 -GBW fac no 1.164 0.609 0.594 0.697 0.429 1.195 4.000 -rcMV small no 1.097 0.557 0.593 0.723 0.497 7.393 0.477 0.816rcMV fac no 1.128 0.573 0.591 0.703 0.526 1.386 0.502 1.015

Table 1: �2 and values of the fitted parameters entering the description of the HERA data. The fit includesthe 252 �

red

data points. The quoted �

2 for �cc̄red

and FL are obtained a posteriori.

with bNf = (11Nc

� 2Nf

)/12⇡. This fudge factor is also included in the rcMV type initial condition in (15).The N

f

-dependent Landau pole is obtained by imposing ↵s(p2 = M

2

Z) = 0.1185 at the scale of the Z mass[63] and continuity of ↵s at the flavour thresholds, using mc = 1.3 GeV and mb = 4.5 GeV. To regularisethe infrared behaviour, we have decided to freeze ↵s at a value ↵

sat

= 1 and we have checked explicitly thatreducing this down to, for example, 0.7 does not a↵ect the fit in any significant manner.

Note that we do not include any form of resummation or matching for ln 1/r2 > Y , as introduced in[50], in these initial conditions. One of the reasons for not doing so is that the extra factor in the initialcondition can always be reabsorbed in a re-parametrisation. Furthermore, a proper matching at small dipolesizes, suited for phenomenological studies, would require a careful treatment of the small-dipole region. Inthat respect, the resummed BK evolution is expected to perform a better job than a fixed matching witha fixed asymptotic behaviour. We leave a better treatment, e.g. a genuine matching to DGLAP evolution,for future work.

Rapidity evolution. Of course this is determined by the resummed BK equation given in (9). Here, we againconsider two separate cases, one in which the evolution resums only the leading double logarithms and onein which it also includes the single ones.

From the dipole amplitude to observables. Once we have the dipole amplitude for all rapidities and dipolesizes, we use the standard dipole formalism to obtain the physical observables:

�

�⇤pL,T(Q

2

, x) = 2⇡R2

p

X

f

Zd2r

Z1

0

dz�� (f)

L,T(r, z;Q2)��2T (ln 1/x̃f , r), (17)

where the transverse and longitudinal virtual photon wavefunctions read

�� (f)L

(r, z;Q2)��2 = e

2

q↵

em

Nc

2⇡2

4Q2

z

2(1� z)2K2

0

(rQ̄f ), (18)

�� (f)T

(r, z;Q2)��2 = e

2

q↵

em

Nc

2⇡2

�⇥z

2 + (1� z)2⇤Q̄

2

fK2

1

(rQ̄f ) +m

2

fK2

0

(rQ̄f ) . (19)

In the above we have introduced the customary notation Q̄

2

f = z(1� z)Q2 +m

2

f , x̃f = x(1 + 4m2

f/Q2), and

we have assumed a uniform distribution over a disk of radius Rp in impact parameter space. The sum in (17)runs over all quark flavours and we will include the contributions from light quarks with mu,d,s = 100 MeVas well as from the charm quark with mc = 1.3 GeV. From the longitudinal and transverse cross-sections,

9

init RC sing. �

2

/npts for Q2

max

cdt. schm logs 50 100 200 400GBW small yes 1.135 1.172 1.355 1.537GBW fac yes 1.262 1.360 1.654 1.899rcMV small yes 1.126 1.172 1.167 1.158rcMV fac yes 1.222 1.299 1.321 1.317GBW small no 1.121 1.131 1.317 1.487GBW fac no 1.164 1.203 1.421 1.622rcMV small no 1.097 1.128 1.095 1.078rcMV fac no 1.128 1.177 1.150 1.131

Table 2: Evolution of the fit quality when including data at larger Q2 (in GeV2).

we can deduce the reduced cross-section and the longitudinal structure function as

�

red

=Q

2

4⇡2

↵

em

�

�⇤pT

+2(1� y)

1 + (1� y)2�

�⇤pL

�, (20)

F

L

=Q

2

4⇡2

↵

em

�

�⇤pL

. (21)

When the quark masses, the value of the strong coupling at the Z mass and its frozen value in the infraredhave been fixed, we are left with 4 or 5 free parameters according to our choice of initial condition: Rp the“proton radius”, Q

0

the scale separating the dilute and dense regimes, C↵ the fudge factor in the runningcoupling in coordinate space, and p which controls the approach to saturation in the initial condition. Forthe rcMV initial condition, we have the extra parameter CMV.

We have fitted these parameters to the combined HERA measurements of the reduced photon-protoncross-section [38]. Since the BK equation is applicable only at small-x, we have limited ourselves to theregion x 0.01. We note that since Eq. (17) probes dipoles at the rapidity ln 1/x̃f , the exact cut we imposeis x̃c 0.01 since the most constraining cut comes from the charm, the most massive quark we include inour model. Accordingly, our initial condition for the BK evolution corresponds to x̃ = 0.01. Furthermore,since we do not expect the BK equation to capture the full collinear physics, we impose the upper boundQ

2

< Q

2

max

. By default we will use Q

2

max

= 50 GeV2 but we will also give results for extensions to largerQ

2. In the default case we have a total of 252 points included in the fit. We have added the statistical andsystematic uncertainties in quadrature.3

The results of our fits for the 23 = 8 cases, depending on the initial condition, the running couplingprescription and the inclusion or not of single logarithms in the kernel, are presented in Table. 1. The tableincludes the parameter values obtained from fitting the �

red

data and, besides the fit �

2, it also indicatesthe �

2 obtained a posteriori for the latest �cc̄red

[64] and FL [65] measurements. These results deserve a fewimportant comments.

(i) In general, the overall quality of the fit is very good, reaching �

2 per point around 1.1-1.2.(ii) Apart from a few small exceptions (see below), all the parameters take acceptable values of order one.

Note that we have manually bounded p between 0.25 and 4. Whenever it reached the upper limit,larger values only led to minor improvements in the quality of the fit.

(iii) The two initial conditions give similar results, with a slight advantage for the rcMV option, which islikely due to the extra parameter. Note that for a standard MV-type of initial condition T (Y

0

, r) ={1� exp[�(r2Q2

0

/4 [c+ ln(1+ 1/r⇤)])p]}1/p, we have not been able to obtain a �

2 per point below 1.3and the parameters, typically c or p, tend to take unnatural values.

3A more involved treatment of the correlated systematic uncertainties leads to similar results with slightly worse �

2 perpoints (about 0.04).

10

Backup slides



Resummation of double logs in BKExtend DLA to match BK

❖ Let 2T̃ (Y, ⇢1) ! T̃13 + T̃23

❖ Restore dipole kernel d⇢1e�(⇢�⇢1) =

dz213 z212

z413! 1

⇡

dz23 z212

z213z223

❖ Insert virtual term , non-linear term, remove cutoffs�T̃12

❖ Replace argument in argument of DLA kernel to switch off resummation for small daughter dipoles

⇢� ⇢1 = lnz213z212

!

s

lnz213z212

lnz223z212

⌘p

L13L23


ω-shift❖ In DLA need x+ ordering → p— ordering

❖ Proper variable Y � = lnp�max

p�= ln

p+

p+min

+ ln z212

Q2

0

= Y � ⇢

❖ Double Mellin representation of BFKL solution

T12 =

Zd�

2⇡i

d!

2⇡i

T̄ 0(� � !)

! � ↵̄s�0(� � !)e!Y (z212Q

20)

�

⇒

⇒

! =↵̄s

1� � + ! Leads to √ seen earlier

T12 =

Zd⇠

2⇡i

d!

2⇡i

T̄ 0(⇠)

! � ↵̄s�0(⇠)e!Y �

(z212Q20)

⇠


Large single logs from large dipoles❖ Easily seen in BFKL, “some effort” to uncover in BK

dT12

dY=

↵̄2s

8⇡2

Zd2z3 d2z4

z434

� 2 +

z213z224 + z214z

223 � 4z212z

234

z213z224 � z214z

223

lnz213z

224

z214z223

�T34

❖ Ordered dipoles z14 ' z24 ' z34 � z13 ' z23 � z12

KgSL ' �6� cos

2 �

12

↵̄2s

⇡2

z21z23z

44

! �11

24

↵̄2s

⇡2

z21z23z

44

.

�T (z12)

�Y

��g

SL

' �11

12↵̄2sz

212

Z 1

z212

dz24z44

lnz24z212

T (z4)

❖ Dominates the LO term

Numerical solution (Balitsky prescirption)


10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

-5 0 5 10 15 20 25

T(ρ

,Y)

ρ=log(1/r2)

DLA+SL resum, β0=0.72, Balitsky

Y=0Y=4Y=8Y=12Y=16

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

dlo

g[Q

2 s(Y

)]/d

Y

Y

speed, β0=0.72, Balitsky

LO

DLA at NLO

DLA resum

DLA+SL resum

Some fitting expressions


T (Y0, r) =

⇢1� exp

�✓r2Q2

s0

4

◆p��1/p

T (Y0, r) =

⇢1� exp

�✓r2Q2

s0

4

↵̄s(CMVr)

1 + log

✓↵̄sat

↵̄s(CMVr)

◆�◆p��1/p

GBW

rcMV

RC (plus freezing)↵s(r) =1

bNf ln⇥4C2

↵/(r2⇤2

Nf)⇤

Fit


10-1

100

101

102

10-7 10-6 10-5 10-4 10-3 10-2

0.045

0.0650.085

0.1

0.150.20.25

0.350.4

0.50.65

0.851.21.5

22.7

3.54.5

6.58.5

101215182227

3545

mv initial condition

ml=0.10 GeVmc=1.3 GeV

r2/npts = 1.13

F 2 (r

esca

led)

x

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