< x ” X ¢4 ; c" e8 ý P ê s” X ¢ V ê s„ ÆX c l “ Ó Þ; c 6 ” X ¢ Factorization Theorem

Quantum Chromodynamics Factorization Theorem in Limit of the Large x

Chul Kim

School of Liberal Arts, Seoul National University of Science and Technology, Seoul 139-743, Korea (Received 30 June 2015 : revised 15 July 2015 : accepted 15 July 2015)

The quantum chromodynamics (QCD) factorizatin theorem is considered for high-energy scat- tering in the limit of large x, with the main focus being deep inelastic scattering (DIS). Using the extensible parton distribution function (PDF) for large x, we derive the QCD factorization theorem for DIS. The PDF in the limited x → 1 consists of collinear and soft interactions, which are not factrorizable further because subleading spectator soft-collinear interactions are indispensable. This fact also indicates that the PDF at large x should be suppressed by some powers of (1 − x). With the extensible PDF, we considered the factorization theorem for the Drell-Yan process in the limit of large x, which might break down because of the nonfactorizable soft interactions.

PACS numbers: 12.38.Bx, 12.39.St, 13.85.-t

Keywords: QCD factorization, Large x, Perturbative calculation, Effective field theory

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PACS numbers: 12.38.Bx, 12.39.St, 13.85.-t

Keywords: QCD factorization, large x, [O1lx>íߖ, Ä»´ò©œ s:r

∗E-mail: [email protected]

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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p^µ= ¯n · pn^µ

2 + p^µ_⊥+ n · pn¯^µ

2 = O(Q) + O(Qλ) + O(Qλ²) (10)

#

Œl"f λ ∼ p⊥/¯n · p  O(1)HŒ•“rÉB>hús. s\¦

Ö

¸6 xôÇ [jl 0pu“Ér SCETü< ° “úÉr Ä»´ò©œ s:r`¦ +‹"f ì

r$3½+É M: Åҁָ6 x÷&HX<, àÔ0AÛ¼àÔ\ Ér„>hü< 1lx 1

p

x . "f, {9 H —:rs n-collinear ©œÜ¼–Ð ¬¹



÷&€, „lÀÓ J^µ\¦ :ŸxK ~½Óئ÷&H—:r“Ér x → 1ôÇ

>

\"f ¯n−collinear {9–Ð ³ð‰³÷&&#Q ½+É כ s.



"f, full QCD\"f_ „lÀӍH ©œ_ hard ©œ  

ñŒ•6 x`¦ —¸¿º &hìrK!Q2; Êê (integrating out), SCET

\

¦ :ŸxK 6£§õ ° sú ³ð‰&³|¨cú e”.

J^µ = ¯qγ^µq = C(Q², µ) ¯ξ_n¯W_¯nY˜_n¯^†γ_⊥^µY_nW_n^†ξ_n+ O(λ) (11)

#

Œl"f, γ⊥^µ = g^µν_⊥ γ_ν s 9 ξn(¯n)H n(¯n)-collinear3$ß¼ ©œ s

. Collinear ]4’H‚ (Wilson line) Wn [8]“Ér

Wn(x) = P exph ig

Z x

−∞

ds¯n · An(s¯n^µ)i

(12)

–

Ð ÅÒ#Qt 9, n−collinear sü@_ Ér îr1lx`¦ HÂÒìr

\

"f λ_ ܼ>pu ú\"f n−collinear /JåÀғ:r_ 4Ÿ¤ õ

&

ñ`¦ã¼|9#Q ·p כ s. #Œl"f ‘P’Hâ–Рצ[j¹¡§ (path ordering)`¦_pôÇ. ¢¸Ér collinear ]4’H‚ W_n¯H nõ

¯

n\¦´úÀDKܼ–Ð+‹ ~1> %3#Qè­q ú e”. Ref. 9\"f ˜Ð“



ü< °ú s λ_ ܼ>pu ú\"f soft ©œ ñŒ•6 x`¦ collinear



©œ\"f ìro½+É Ãº e”. s M:,
š¸Hכ s soft ]4’H‚ Ü

¼–Ð sH6£§ °õú s ³ð‰&³)a.

Y_n(x) = P exph ig

Z x

−∞

dsn · A_s(sn^µ)i

(13) Y˜_n(x) = ¯P exph

−ig Z +∞

x

dsn · A_n(s¯n^µ)i

(14)

#

Œl"f ¯P H ìøÍâ–Рצ[j¹¡§ (anti-path ordering)`¦ >pwôÇ



. ˜Ynõ Yn_ â–Ð ØÔ> ³ðrH†d[20]\ Ä»_ #Œ.

Eq. (11)\"f ]4’H>úH αs_ !QFKܼ>pu út

C(Q², µ) = 1 +αsCF

4π



− ln² µ²

Q² − 3 ln µ²

Q² − 8 + π² 6

 (15)

–

Ð ÅÒ#Q” [10].

Eq. (3)ܼ–Рئµ1ÏK x → 1 ôÇ>\"f ½¨›¸ †<Êú F1`¦ SCET\¦ :ŸxK ³ð‰&³ €,

F1(x, , Q²) = −g_⊥^µν

2 Wµν = − 1 4π

X

X

(2π)⁴δ(q + P − pX)hN |J_⊥µ^† |XihX|J_⊥^µ|N i

= −4π³X

X

δ(q + P − pX)|C(Q², µ)|²hN | ¯ΨnY_n^†γ_⊥^µY˜n¯Ψ¯n|XihX| ¯Ψn¯Y˜_n¯^†γ_µ^⊥Y˜nΨn|N i (16)

#

Œl"f Ψⁿ ≡ W_n^†ξn–Ð >st Ô¦“ n−collinear 3$ß¼



©œs. x → 1“ âĺ, Ùþ˜ N_ îr1lx|¾Ó_ @/ÂÒìr`¦ {9



 H —:rs 4RH sכÙ¼–Ð Ùþ˜_
 QtH @/ Â

Òìr_ \-t\¦ { 9> |¨c sכ. "f, îr1lx ›¸| \



 þj7áx ©œI\ n-collinear {9 ”>rF H כ “Ér Ô¦

 Ù¼–Ð À» d”\"f |Xi = |X^n¯i ⊗ |Xsi–Ð ÅÒ#Q|9 כ s



.

¯

n-collinear ÂÒìr“Ér Aü< °ú s Z>>h_ jet functionܼ

–

Ð ¬¹ Hכ s 0px .

X

X_¯n

h0|Ψ^α_n¯|Xn¯ihXn¯| ¯Ψ^β_n¯|0i =

Z d⁴pX_¯n

(2π)³ 6 ¯n

2n·p¯ X_n¯Jn¯(pX_n¯)δ^αβ. (17)

#

Œl"f α, βHҐo¾ú˜\ @/ôÇ €ªœú (quantum number)\¦

·p. ܼ>pu ú\"f s jet function“Ér J_n⁽⁰⁾_¯ (p) = δ(p²)ܼ–Ð ÅÒ#Q”.

î

r1lx|¾Ó ˜Ð”>rõ ,|9 0A Ä»•¸ (on-shell degrees of freedom)\¦ “¦9 € ¯n-collinear îr1lx|¾Óõ soft îr1lx|¾Ó

“

Ér 1 − x ∼ O(λ²)–Ð ÑütM: 6£§ °õú s Û¼H{9aA Hכ

`

¦·ú˜ ú e”.

p¯n = (p⁺_¯n, p_n¯⁻, p^⊥_n¯) = (λ², 1, λ),

ps = (p⁺_s, p⁻_s, p^⊥_s) = (λ², λ², λ²) (18)

#

Œl"f p^¯n≡ pX_n¯, ps≡ pX_ss“¦, p⁺≡ ¯n · p, p⁻≡ n · p–Ð

¿

º%3 (·ú¡Ü¼–Е¸ s ³ðl\¦>5ÅqÖ¸6 x½+É כ s.). 

"

f, Eq. (16)\"f ÚÔYUsàÔ d¦"\f_ 4Sq †<ÊúH

δ(q + P − pn¯− ps) = 2δ(−Q + P⁺− p⁺_¯n − p⁺_s)δ(Q − p_n⁻_¯ − p⁻_s)δ⁽²⁾(p^⊥_n¯ + p^⊥_s)

∼ 2δ(Q1 − x

x − p⁺_n¯ − p⁺_s)δ(Q − p_n⁻_¯)δ⁽²⁾(p^⊥_¯n) (19)

–

Ð éߖíHoa). #Œl"f ›'aÏ|1¹¾Ó x\¦Ÿí†<Ê t ·ú§H îr1lx

|

¾Ó_ $íìr\ @/K"fH[jl 0pu`¦ :Ÿx #Œ Œ•> l#Œ 



HÂÒìr“ÉrÁºr %i. "f, ÚÔYUsàÔ d¦\"fH H x ô

Ç>\"f †½Ó©œ p⁻n¯ = Q, p^⊥_n¯ = 0ܼ–Ð jþtú e”6£§\ Ä»_

#Œ.

Eq. (17)`¦ Eq. (16)\ @/{9ôÇ +' Eq. (19)\¦&h6 x #Œ

&

ño €,

F1(x, Q²) = H(Q², µ)QX

Xs

Jn¯(p⁺_n¯ = Q1 − x

x − p⁺_s; Q)hN | ¯ΨnY_n^†Y˜n¯|Xsi6 ¯n

2hXs| ˜Y_n¯^†Y˜nΨn|N i

= H(Q², µ)Q² x

Z 1 x

dyJn¯(Qy − x

x ; Q)hN | ¯ΨnY_n^†Y˜n¯

6 ¯n

2δ(yP⁺− P⁺− i∂⁺) ˜Y_n¯^†Y˜nΨn|N i

= H(Q², µ) Z 1

x

dy y

J¯n¯(1 − x

y; Q)hN | ¯ΨnY_n^†Y˜¯n

6 ¯n

2δ(yP⁺− P⁺− i∂⁺) ˜Y_n¯^†Y˜nΨn|N i (20)

–

Ð jþt ú e”. #Œl"f H(Q², µ) = |C(Q², µ)|²s. s d”`¦ Ä»•¸½+É M:, ¿ºP: צ"\f ¢-a„ ›'a> (com-

pleteness relation) P

X_s|XsihXs| = 1`¦ 6&h x %i“¦, R dyδ(y−· · · )\¦ƒíߖ s\ ¶š{ú9 %i. P<sup>+</sup>ü< i∂<sup>+</sup>H collinear©œõ soft ©œ\ yŒ•yŒ• &h6 x&÷H îr1lxÓ`¾|¦ã¼|9#Q

?

/Hpìrƒíߖs. t}Œ• d”\"f ½©o)ajet func- tion“Ér "é¶A jet functionõ 6£§õ ° “úÉr›'a>\¦”.

J¯n¯ = y

xQ²Jn¯ ∼ Q²Jn¯ (21)



6£§©œ\"f [jy ¶˜(úR˜Ð’xtëߖ, Eq. (20)\"f Ùþ˜ N s_ 'Ÿ§>= כ¹™è (matrix element)H—:r ìrŸí †<Êú

 H y (¢¸H x)\¦ tu´ M:_ ‰XS©œ)a ð‰³&³Ü¼–Ð “¦9|¨c Ã

º e”.

III. ½  ÊX ê s  Ö «” X ¢  Ê ] Ø Ä Z ؃ º ] K ¤• ¤õ u § T “ Ó Þ” X ¢ QCD factorization theorem

Ùþ

˜–ÐÂÒ' :£¤&ñôÇ —:rs Ùþ˜\ @/ôÇ \-t qÖ¦ x\¦ t“¦ {9 #Œ “¦\-t ©œ ñ Œ•6 x`¦ ½+É SX‰Ò¦`¦

?/H —:r ìrŸí †ÊÃ<ºH full QCD\"f 6£§õ °ú s

&

ñ_)a [21].

f_q/N(x, µ) =

Z dz₋

4π e^{−ixP+z−/2}hN (P )|¯qn(z₋ 2 )6 ¯n

2P exph ig

Z z−/2 0

d¯z⁰n · A(¯¯ z⁰)i

qn(0)|N (P )i (22)

= hN (P )|¯q_n6 ¯n

2δ(xP₊− ¯n · iD)q_n|N (P )i (23)

#

Œl"f qn“Ér n-~½Ó†¾Óܼ–Ð collinearôÇ 3$ß¼©œs 9 >st Ô

¦`¦ 0AK"f ¯n-~½Ó†¾Ó`¦ ¹¡§f”sH s>t /BN ]4

’

H‚s ¶úš{9÷&%3. ¿ºP: צ“Ér ÉÓo#Q (Fourier) ¨8Š s

Êê îr1lxÓ /¾|BNçߖ\"f ³ð‰³ô&Ç כ s. /åJÀғ:r\ @/ôÇ 

—

:r ìrŸí †<Êú•¸ sü< Ä»ôÇ ~½Ód”ܼ–Ð &ñ_|¨cú e”.

à

Ô0AÛ¼àÔ (¢¸H λ)\ @/ôÇ Ü¼>pu ú\"f SCET\ B gA(matching) #Œ s\¦³ð‰&³ €,

f_q/N(x, µ) = hN (P )| ¯ξn

6 ¯n

2δ(xP+− ¯n · iDn− Wnn · iD¯ sW_n^†) ξn|N (P )i

= hN (P )| ¯ξnWn

6 ¯n

2δ(xP+− P+− ¯n · iDs) W_n^†ξn|N (P )i

= hN (P )| ¯ξ_nW_nY˜_n¯ 6 ¯n

2δ(xP₊− P+− i∂+) ˜Y_n¯^†W_n^†ξ_n|N (P )i

= hN (P )| ¯ξ_n⁽⁰⁾W_n⁽⁰⁾Y_n^†Y˜_¯n6 ¯n

2 δ(xP₊− P₊− i∂₊) ˜Y_n¯^†Y_nW_n^(0)†ξ⁽⁰⁾_n |N (P )i (24)

–

Ð jþtú e”. 'Í P: צ\"f full QCD_ /BN pìrƒ í

ߖ (covariant derivative) ¯n · iDH SCET\"f

¯

n · iD = ¯n · iD_n+ W_nn · iD¯ _sW_n^† (25)

–

Ð ³ð‰&³½+É Ãº e”. #Œl"f ¯n · iDⁿ“Ér n−collinear /BN

ƒ

íߖs 9, ¯n · iDsH soft ©œ\ @/ôÇ כ s. À» d”“Ér SCET Õª|½Ótîߖ x9 ƒíߖ\¦ „>h½+É M:, λ_ —¸ŽHyŒ• y

Œ

•_ ú\"f >st Ô¦Ü¼–Ð ëߖ[þtl 0A #Œ collinear



©œ`¦ &h]X > &ñ_†<Êܼ–Ð+‹ %3H ³ð‰&³d”s [22]. p ì

rƒíߖ Pü< i∂“Ér collinear©œõ soft ©œ\ yŒ•yŒ• &h6 x÷&



H כ ܼ–Ð [P, φs] = [i∂, φ_n] = 0`¦–7ßëá¤ôÇ. (φn,sH e” _

_ collinear(soft) ©œs.) Eq. (24)_ ¿ºP: צõ [j



P: צ\"f e”__ †<Êú f\ @/ #Œ †½Ó1pxd” f (¯n·iDn) = Wnf (P+)W_n^†, f (¯n · iDs) = Yn¯f (i∂+)Y_n¯^†\¦ 6 x %i.

Eq. (24)_ t}Œ• õH λ _ ܼ>pu ú\"f n-collinear



©œÜ¼–ÐÂÒ' soft ©œ ñ Œ•6 x`¦ ìroK ·p כ s [9]. Õª



õ, collinear ©œ[þt“Érܼ>pu ú\"f 8 s©œ soft ©œ ñ



Œ

•6 x`¦ t ·ú§>)a.

Eq. (24)_ t}Œ• ³ð‰&³d“”Ér full QCD–Ð ÂÒ' ƒÛ¼ XO

> Ä»•¸÷&H SCET\"f_ —r ì:rŸí †<Êú_ &ñ_s 9, s

–ÐÂÒ' x → 1“ ôÇ>t 1lx&Üh¼–ÐSX‰©œ½+É Ãº e”.

Ä

º‚ Eq. (24)\"f x_ —¸ŽH#30A\¦“¦9½+É âĺ (7£¤, x ∼ O(1) & 1 − x ∼ O(1)“ âĺ), 4Sq †<Êú_ argumentH

δ(xP+− P+− i∂+) = δ(xP+− P+) + O(i∂+)–Ð „>h|¨c Ã

º e”“¦, ¿ºP: †½Ó“Érܼ>up ú\"f Áºr½+É Ãº e”. 



"f, s âĺ —:r ìrŸí †<ÊúH e”y ¸ú˜ ·ú˜9” 6£§õ

° ú

 “Ér³ð‰&³d”ܼ–Ð r) ôÇ.

f_q/N(x, µ) ∼ hN (P )| ¯ΨnY_n^†Y˜n¯

6 ¯n

2δ(xP+− P+) ˜Y_n¯^†YnΨn|N (P )i

= hN (P )| ¯Ψn

6 ¯n

2δ(xP+− P+)Ψn|N (P )i (26)

#

Œl"f Ψⁿ = W_n^†ξns 9, ¿ºP: d”\"f ]4’H‚_ Ä»m '

o (unitary)›'a>d” Y Y^† = Y^†Y = 1`¦+‹"f —¸ŽH soft



©œ_ l#Œ\¦]j %i.

ë

ߖ{9 x → 1 ôÇ>\ [þtQ"#f> ÷&€, Ö¸$ío)a n−collinear—:r“ Ψn“ÉrÙþ˜_ _ —¸ŽH\-t\¦ t

Ù¼–Ð P+Ψ_n = P₊Ψ_ns ÷&“¦, s âĺ 4Sq †<Êú_ argumentH (1 − x)P+ + i∂+–Ð jþt º eà ”> )a. 

"

f, ĺo H x ôÇ>\"f_ ½¨›¸ †<Êú\¦ “¦9½+É âĺ (1 − x)P₊∼ i∂+–Ð [jl 0pus ÷&#Q —:r ìrŸí †<Êú ?/

\

"f soft ©œ`¦ ]j½+É Ãº \O> ÷&“¦ Eq. (24)_ t}Œ•

³

ð‰&³d”`¦ :ŸxK —:r ìrŸí †<Êú\¦“¦9Këߖ Hכ s



.



"f, Eq. (24)_ ›'a>d`”¦ Eq. (20)\ @/{9K ˜Ð

€

, ĺoH H x ôÇ>\"f_ ½¨›¸ †<Êú F1\ @/ôÇ QCD factorization theorem`¦6£§õ °ú s ½¨½+É Ãº e”.

F1(x, Q²) = H(Q², µ) Z 1

x

dy

y f_q/N(y, µ) ¯Jn¯(1 − x y; Q, µ)

= H(Q², µ) Z 1

x

dy y f_q/N(y

x, µ) ¯Jn¯(1 − y; Q, µ) (27)



²DG, d”8£x òqøÍ$í Øæ[t\"f ½¨›¸ †<ÊúH H x ôÇ>\

"

f hard ©œ ñ Œ•6 x`¦ ìroK ·p Êê, ±ú“Ér \-t\"f

¯

n−collinear ©œ ñ Œ•6 x`¦ ³ð‰&³ H jet functionõ SX‰©œ

)

a³ð‰&³_ —:r ìrŸí †<Êú–Ð ¥¸>h#Q ³ð‰&³½+É Ãº e”. #Œ l

"f —:r ìrŸí †<ÊúH–íéßHôÇ (n−)collinear ©œ ñ Œ•6 x

÷

rëߖ m soft ©œ ñ Œ•6 x•¸ ŸíF‹c½+É Ãº e”6£§\ Ä»_ 

#

Œ ôÇ.

Eq. (27)\"f jet fucntion_ αs_ !QFKܼ>pu ú\"f _

 >íߖ“Ér6£§ °õú s ·ú˜94R e” [23,24].

J (1 − z; Q, µ) = δ(1 − z) +αs

2πCF

(

δ(1 − z)

"

7 2 −π²

2 +3 2ln µ²

Q² + ln² µ² Q²

#

− 1 (1 − z)₊

"

2 ln µ² Q²+3

2

#

+ 2 ln(1 − z) 1 − z

!

+

)

. (28)

#

Œl"f (f(z))+“Ér e¦QÛ¼ ìrŸí †<Êú\¦
?/ 9, Õª &ñ _

H6£§õ °ú .

 f (x)

+

= lim

β→0

"

f (x)θ(1−x−β)−δ(1−x−β) Z 1−β

0

dyf (y)

#

(29)



"f, Eq. (28)`¦ :ŸxÇ jet function_ô  s©œ "é¶ (anomalous dimension)“Ér

γ_J(z, µ) = α_sC_F π

h2 ln µ² Q² +3

2

δ(1 − z) − 2 (1 − z)+

i (30)

–

Ð ÅÒ#Q”. 0A\"f ½¨ôÇ s©œ "鶓Ér 6£§õ °ú “Ér F

½

©o ÕªÒ¨~½Ó&ñd” (renormalization group equation)`¦ ë

ߖ7ᤠH°úכs.

d

d ln µJ (x, µ) = Z 1

x

dz

z γJ(z, µ)J (x

z, µ) (31) Hard function H(Q², µ)_ s©œ "鶓Ér

γH(µ) = 1 H(Q², µ)

d

d ln µH(Q², µ) = −αsCF

π

 2 ln µ²

Q²+3 (32)

–

Ð ÅÒ#Q”.

—:r ìrŸí †<Êú_ s©œ "¶“éÉr ¸ú˜ ·ú˜9” ü< °ú s Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)& V,

 (kernel) [25–27]–Ð ·ú˜94R e”. H x ôÇ>\"f•¸ s

\

¦ëߖ7ᤠHt ¶ú˜(R˜Ðl 0A #Œ —:r "¶é\"f —:r ìr

Ÿ

í †<Êú\¦ “¦9K ˜Ðl–Ð . F½©o Û¼H{9aA 'Ÿ 1

lx (renormalization scaling behavior)“Ér š¸–Ðt ƒíߖ

\

 _”>r½+É  sכÙ¼–Ð —:r "¶é_ s©œ "鶓Ér z´]j y©œ{9



 "é¶_ —:r ìrŸí †<Êúü< °ú .

f_q/q(x, µ) = hq(p)| ¯Ψ_nY_n^†Y˜_n¯6 ¯n

2δ((1 − x)p₊+ i∂₊) ˜Y_n¯^†Y_nΨ_n|q(p)i (33)

= 1

p₊hq(p)| ¯Ψn|0i6 ¯n

2h0|Ψn|q(p)i · 1

N_ch0|Y_n^†Y˜¯n

6 ¯n

2δ((1 − x) +i∂+

p₊) ˜Y_¯n^†Yn|0i (34)

≡ C(p+, µ) · S(1 − x; p+, µ)

¿

º P: ³ð‰&³d”`¦ ã¼|#9Q è­q M:, —:r "é¶\"f collinearü< soft ©œs λ\ @/ôÇ Ü¼>pu ú\"f 8 s©œ



©œ ñ Œ•6 x t ·ú§HH `כ¦“¦9 #Œ s[þts\ &h ]X

y |0ih0|\¦¶úš{9 %i.

αs_ !QFK ܼ>pu út_ collinear functionõ soft function_ >íߖ°úכ“Ér

C(p+, µ) = 1 +αsCF

π

 1

_UV − 1

_IR

 1

_UV + 1 + ln µ p+

−αsCF

4π

 1

_UV − 1

_IR



(35) S(1 − x; p+, µ) = δ(1 − x) +αsCF

π

 1

_UV − 1

_IR

h− 1

_UV − ln µ p+



δ(1 − x) + 1 (1 − x)+

i

(36)

–

Ð ÅÒ#Q”. "f, —:r "¶é_ —:r ìrŸí †<ÊúH

f_q/q(x, µ) = C(p+, µ) · S(1 − x; p+, µ)

= δ(1 − x) +α_sC_F 2π

 1

_UV − 1

_IR

h3

2δ(1 − x) + 2 (1 − x)+

i (37)

–

Ð ÅÒ#Qt“¦, s\ @/ôÇ sœ ©"鶕¸ &ñ½© DGLAP &V,

° ú

כ`¦”.

γf(x, µ) = P_q/q(x → 1) =αsCF

π h3

2δ(1 − x) + 2 (1 − x)+

i (38)

¢

¸ôÇ, Eqs. (30), (32), (38)\¦ :ŸxK Eq. (27)\"f ÅÒ#Q” Ó

ü

to&h“a8'›£¤°úכ“ ½¨›¸ †<Êú F¹s F½©o Û¼H{9 µ\ _

”>r t ·ú§“¦ d/(d ln µ)F1 = 0 `¦–7ßëᤠ“¦ e”6£§`¦ ~1

>

SX‰“½+É Ãº e”.

t

FKt 7H_)a \¦ “¦9 €, SCET\¦ :ŸxK"f ±ú

“ É

r\-t\"f_ "f–Ð Ér îr1lx›¸| `¦” ©œ[þt`¦$í /

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