hbbm;?m qq`fb?qtqmjgbm:2qk2i`v m/s?vbb+b-rr ......t`2/b+ibqmb7q`b2p2` ht`q+2bb2b ig>*x ⇒...

$: hbBM;?m qQ`Fb?QTQMJGBM:2QK2i`v M/S?vbB+b-RR ......T`2/B+iBQMb7Q`b2p2` HT`Q+2bb2b iG>*X ⇒ L22/Q7T`2+Bb2S.6/2i2`KBM iBQM M/mM+2`i BMiv2biBK i2X N q?vJGBMS.6b/2i2`KBM iBQM\ ÇS.6b `22bb2MiB$
JG M/ S`iQM .Bbi`B#miBQM 6mM+iBQMb

ai27MQ *``xxhbBM;?m qQ`Fb?QT QM JG BM :2QK2i`v M/ S?vbB+b- RR@R8 CmM2 kyR3- aMv

1m`QT2M P`;MBxiBQM 7Q` Lm+H2` _2b2`+? U*1_LV

+FMQrH2/;2K2Mi, h?Bb T`QD2+i ?b `2+2Bp2/ 7mM/BM; 7`QK >A**lS 1_* *QMbQHB/iQ`;`Mi UeR98ddV M/ #v i?2 1m`QT2M lMBQMǶb >Q`BxQM kyky `2b2`+? M/ BMMQpiBQMT`Q;`KK2 mM/2` ;`Mi ;`22K2Mi MQX d9yyyeX

PDFN 3Machine Learning • PDFs • QCD

PmiHBM2

Ç aMTb?Qi Q7 JG BM i?2Q`2iB+H ?B;?@2M2`;v T?vbB+bÇ AMi`Q/m+iBQM iQ T`iQM /Bbi`B#miBQM 7mM+iBQMb US.6bVÇ JG TTHB2/ iQ S.6 /2i2`KBMiBQM

k

q?i Bb JG BM >1S@S> T?vbB+b\

hQ/vǶb HM/b+T2 +M #2 ;`QmT2/ BMiQ irQ 2Mb2K#H2b,

:`QmT UN8W Q7 TTHB+iBQMbV

*QKTmiiBQMH i2+?MB[m2b M/ iQQHb

Ç /pM+2/ MmK2`B+H K2i?Q/bM/ TTHB+iBQMb

Ç JQMi2 *`HQ 2p2Mi ;2M2`iQ`bÇ >B;?2` Q`/2`b +Q``2+iBQMbÇ *QKTmi2` H;2#` i2+?MB[m2b

h?2b2 `2 i?2 KQbi TQTmH`+QKTmiiBQMH >1S@h> T?vbB+biQTB+b r?B+? Kv BM+Hm/2 KQ/2`MJG i2+?MB[m2b QM iQT Q7 /pM+2/+QKTmiiBQMH T?vbB+bX

j


hQ/vǶb HM/b+T2 +M #2 ;`QmT2/ BMiQ irQ 2Mb2K#H2b,

:`QmT UN8W Q7 TTHB+iBQMbV

*QKTmiiBQMH i2+?MB[m2b M/ iQQHb

Ç /pM+2/ MmK2`B+H K2i?Q/bM/ TTHB+iBQMb

Ç JQMi2 *`HQ 2p2Mi ;2M2`iQ`bÇ >B;?2` Q`/2`b +Q``2+iBQMbÇ *QKTmi2` H;2#` i2+?MB[m2b

h?2b2 `2 i?2 KQbi TQTmH`+QKTmiiBQMH >1S@h> T?vbB+biQTB+b r?B+? Kv BM+Hm/2 KQ/2`MJG i2+?MB[m2b QM iQT Q7 /pM+2/+QKTmiiBQMH T?vbB+bX

:`QmT " U8W Q7 TTHB+iBQMbV

JG i2+?MB[m2b #b2/ QM /iX

Ç _2;`2bbBQM M/ +HbbB+iBQMUbmT2`pBb2/ H2`MBM;V

Ç h2+?MB[m2b 7Q` mM+2`iBMivT`QT;iBQM M/ +QK#BMiBQM

Ç 1tT2`BK2MiH Ki?2KiB+b mbBM;JG QTiBKBxiBQM

lbmHHv i?Bb ;`QmT Bb +HQb2` iQ>1S@1sS TTHB+iBQMbX 1bv iQ M/?v#`B/ T`QD2+ib +Qp2`BM;2tT2`BK2MiH M/ i?2Q`2iB+H T?vbB+bX

j


aQK2 2tKTH2b 7`QK :`QmT ,

Ç LmK2`B+H K2i?Q/b M/ i2+?MB[m2b 7Q` L@HQQT BMi2;`Hb-Ç JQMi2 *`HQ i2+?MB[m2b M/ bm#i`+iBQM b+?2K2b

aQK2 2tKTH2b 7`QK :`QmT ",

Ç _2;`2bbBQM- LL KQ/2Hb- `2r2B;?iBM;, SìQM /Bbi`B#miBQM 7mM+iBQMb-7`;K2MiiBQM 7mM+iBQMb- JQMi2 *`HQ imM2b- JALGP `2r2B;?iBM;X

ULLS.6 `sBp,ReRkXyR88R- `sBp,RdyeXydy9N- `sBp,Rey8Xye8R8- a* 2i HX R3y8XyN388VÇ *HbbB+iBQM, .22T *LL D2i /Bb+`BKBMiBQMX UEQKBbF2 2i HX `sBp,ReRkXyR88RV

Ç lM+2ìBMiv 2biBKiBQM@+QK#BMiBQM, S.69G>*R8 iQQHb- ?B;?2`@Q`/2`mM+2ìBMiv KQ/2HBM;X

US.69G>* 2i HX `sBp,ReRkXyR88R- a* `sBp,Rdy9Xyy9dRV

Ç 1tT2`BK2MiH Ki?2KiB+b, HQQT 2pHmiBQM i2+?MB[m2b- M2r KQ/2Hb#b2/ QM T?vbB+b

U.X E`2~ 2i HX `sBp,RdRkXyd83R- `sBp,R3y9Xydde3V

9

*b2 bim/v, i?2 T`QiQM bi`m+im`2 /2i2`KBMiBQM

.m`BM; i?2 Hbi v2`b i?2 S.6 +QKKmMBiv ?p2 Tm#HBb?2/ BMMQpiBp2`2bmHib BM >1S@S> mbBM; JG K2i?Q/b,

Ç LLS.6, U`sBp,ReRkXyR88R- RdyeXydy9N- Rey8Xye8R8V

Ç /2i2`KBMiBQM Q7 i?2 BMi2`MH bi`m+im`2 Q7 +QKTQbBi2 T`iB+H2b- 2X;XTQH`Bx2/ M/ mMTQH`Bx2/ T`QiQM M/ 7`;K2MiiBQM 7mM+iBQMbX

Ç S.69G>* `2+QKK2M/iBQM, U`sBp,ReRkXyR88R- R8y9Xye98N- R8y9XyedjeV

Ç S.6 +QK#BMiBQMÇ S.6 BM7Q`KiBQM QTiBKBxiBQMf+QKT`2bbBQM

AM i?2 M2ti bHB/2b r2 b?Qr [mB+F BMi`Q/m+iBQM #Qmi JG 7Q` S.6/2i2`KBMiBQMX

8

AMi`Q/m+iBQM iQ S.6b

SìQM /2MbBiv 7mM+iBQMb

h?2 TìQM KQ/2H rb BMi`Q/m+2/ #v 62vMKM BM RNeN BM Q`/2` iQ+?`+i2`Bx2 ?/`QMb U2X;X T`QiQMb M/ M2mi`QMbV BM Z*. T`Q+2bb2b M/BMi2`+iBQMb BM ?B;? 2M2`;v TìB+H2 +QHHBbBQMbX

SìQMb `2 [m`Fb M/ ;HmQMb +?`+i2`Bx2/ #v T`Q##BHBiv /2MbBiv7mM+iBQMb Q7 Bib Mm+H2QM KQK2MimKX

e


Ç S.6b `2 2bb2MiBH 7Q` `2HBbiB+ +QKTmiiBQM Q7 Mv T`iB+H2 T?vbB+bQ#b2`p#H2- σ- i?MFb iQ i?2 7+iQ`BxiBQM i?2Q`2K

σ = σ ⊗ 7,

r?2`2 i?2 2H2K2Mi`v ?`/ +`Qbb@b2+iBQM σ Bb +QMpQHmi2/ rBi? 7 i?2 S.6XÇ S.6b `2 MQi +H+mH#H2, `2~2+i MQM@T2`im`#iBp2 T?vbB+b Q7 +QMM2K2MiXÇ S.6b `2 2ti`+i2/ #v +QKT`BM; i?2Q`2iB+H T`2/B+iBQMb iQ `2H /iX

x

3−10 2−10 1−10 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

g/10

vu

vd

d

c

s

u

b

NNPDF3.0 (NNLO)

)2 GeV4=102µxf(x,

LHC typical scale

x

3−10 2−10 1−10 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

g/10

vu

vd

d

u

s

c

b

)2 GeV8

=102µxf(x,

FCC typical scale

d


Ç S.6b `2 M2+2bb`v iQ /2i2`KBM2 i?2Q`2iB+H T`2/B+iBQMb 7Q`bB;MHf#+F;`QmM/ Q7 2tT2`BK2MiH K2bm`2K2MibXÇ 2X;X i?2 >B;;b /Bb+Qp2`v i i?2 G>*,

3

S.6 mM+2ìBMiB2b

S.6 /2i2`KBMiBQM `2[mB`2b b2MbB#H2 2biBKi2 Q7 i?2 mM+2ìBMiv- M/ MQiQMHv i?2 +2Mi`H pHm2- bQ MQi r2HH `2b2`+?2/ iQTB+ BM JGX

*1_L u2HHQr _2TQì 9 UkyReV

S.6 mM+2ìBMiB2b `2 HBKBiBM; 7+iQ` BM i?2 ++m`+v Q7 i?2Q`2iB+HT`2/B+iBQMb 7Q` b2p2`H T`Q+2bb2b i G>*X

⇒ L22/ Q7 T`2+Bb2 S.6 /2i2`KBMiBQM M/ mM+2ìBMiv 2biBKi2X

N

q?v JG BM S.6b /2i2`KBMiBQM\

Ç S.6b `2 2bb2MiBH 7Q` `2HBbiB+ +QKTmiiBQM Q7 ?/`QMB+ T`iB+H2T?vbB+b Q#b2`p#H2- σ- i?MFb iQ i?2 7+iQ`BxiBQM i?2Q`2K- 2X;X BM TT+QHHB/2`,

σs(b,Jks︸︷︷︸

u

) =∑

,#

ˆ R

tKBM

/tR/tk σ,#(tR, tk, b,Jks)︸︷︷︸

s

7(tR,Jks)7#(tk,Jk

s),

r?2`2 i?2 2H2K2Mi`v ?`/ +`Qbb@b2+iBQM σ Bb +QMpQHmi2/ rBi? 7 i?2 S.6XÇ 7B(tR,Jk

s) Bb i?2 S.6 Q7 T`iQM B +``vBM; 7`+iBQM Q7 KQK2MimK t ib+H2 J ⇒ M22/b iQ #2 H2`M2/ 7`QK /iX

Ç *QMbi`BMib +QK2 BM i?2 7Q`K Q7 +QMpQHmiBQMb,

s ⊗ 7 → u

Ç 1tT2`BK2MiH /i TQBMib Bb 8yyy → MQi #B; /i T`Q#H2KÇ .i 7`QK b2p2`H T`Q+2bb M/ 2tT2`BK2Mib Qp2` i?2 Tbi /2+/2b

⇒ /2H rBi? /i BM+QMbBbi2M+B2b

Ry

JG M/ S.6 /2i2`KBMiBQM

h?2 LLS.6 K2i?Q/QHQ;v

h?2 LLS.6 UL2m`H L2irQ`Fb S.6V BKTH2K2Mib i?2 JQMi2 *`HQTT`Q+? iQ i?2 /2i2`KBMiBQM Q7 ;HQ#H S.6 iX q2 T`QTQb2 iQ,

RX `2/m+2 HH bQm`+2b Q7 i?2Q`2iB+H #Bb,Ç MQ t2/ 7mM+iBQMH 7Q`KÇ TQbbB#BHBiv iQ `2T`Q/m+2 MQM@:mbbBM #2?pBQ`

⇒ mb2 L2m`H L2irQ`Fb BMbi2/ Q7 TQHvMQKBHbkX T`QpB/2 b2MbB#H2 2biBKi2 Q7 i?2 mM+2ìBMiv,

Ç mM+2ìBMiB2b 7`QK BMTmi 2tT2`BK2MiH /iÇ KBMBKBxiBQM BM2+B2M+B2b M/ /2;2M2ì2 KBMBKÇ i?2Q`2iB+H mM+2ìBMiB2b

⇒ mb2 J* ìB+BH `2THB+b 7`QK /i- i`BMBM; rBi? : KBMBKBx2`jX h2bi i?2 b2imT i?`Qm;? +HQbm`2 i2bib

RR

1tT2`BK2MiH /i

h?2 iQiH MmK#2` Q7 /i TQBMib 7Q`i?2 /27mHi S.6 /2i2`KBMiBQM Bb

Ç 9Rd8 i GP- 9kN8 i LGP M/9k38 i LLGPX

Ç d T?vbB+H T`Q+2bb2b 7`QK R92tT2`BK2Mib Qp2` jy v2`bU/2H rBi? /i BM+QMbBbi2M+B2bV

Ç 72r /i TQBMib i ?B;? M/ HQrt U/2H rBi? 2ti`TQHiBQMV

Ç `M;2 Q7 8 M/ d Q`/2`b Q7K;MBim/2 T2` S.6 2pHmiBQM`;mK2Mib (t,Zk)

Rk

.:GS 2pQHmiBQM

*M r2 `2/m+2 i?2 S.6 BMTmi bBx2\ u2b- i?MFb iQ .:GS,

7B(tα,Zk) = Γ(Z,Zy)BDαβ7D(tβ ,Zky)

q2 `2KQp2 i?2 Zk /2T2M/2M+2 7`QK S.6 /2i2`KBMiBQM i?MFb iQ i?2.:GS 2pQHmiBQM QT2ìQ` ΓX

7(t,Zk) → 7(t,Zky) := 7(t)

Ç S`2+QKTmi2 i?2 .:GS QT2ìQ` 7Q` HH /i TQBMibÇ TTHv i?2 QT2ìQ` iQ i?2 TìQMB+ +`Qbb b2+iBQMÇ aiQ`2 i?2 `2bmHib M/ T2`7Q`K 7bi +QMpQHmiBQMb

AM LLS.6 i?2Q`2iB+H T`2/B+iBQMb `2 biQ`2/ BM S61G;`B/ i#H2b,

σ =M7∑

B,D

Mt∑

α,β

qBDαβ7B(tα,Zky)7D(tβ ,Zk

y)

Rj

6bi i?2Q`v +QKTmiiBQM

S61G;`B/ U"2ìQM2 2i HX- `sBp,Rey8XykydyV +QMp2ìb BMi2`TQHi2/ r2B;?ii#H2b T`QpB/2/ #v SSG;`B/ BM M 2+B2Mi 7Q`Ki 7Q` S.6 iiBM;- 2X;X

σ =M7∑

B,D

Mt∑

α,β

qBDαβ7B(tα,Zky)7D(tβ ,Zk

y)

r?2`2 ;`B/b `2 T`2@+QMpQHmi2/ rBi? S.6 2pQHmiBQM F2`M2Hb 7`QK S61GXU"2ìQM2 2i HX- `sBp,RjRyXRjN9V

Sm#HB+ +Q/2, ?iiTb,ff;Bi?m#X+QKfM?ìHM/fS61G;`B/ R9

.2MBM; i?2 JG T`Q#H2K

AM +QKT`BbQM iQ ivTB+H JG T`Q#H2K- S.6 i

Ç `2[mB`2b biiBbiB+HHv bQmM/ mM+2`iBMiv 2biBKi2Ç Bb `2;`2bbBQM T`Q#H2K #mi +QKTH2t /2T2M/2M+2 QM S.6bÇ Kmbi biBb7v T?vbB+H +QMbi`BMb,

Ç 7(t) → y 7Q` t → R U+QMiBMmBivVÇ bmK `mH2b,

M7∑

B

ˆ R

y/t t7B(t) = R,

ˆ R

y/t (m(t)− m(t)) = k

ˆ R

y/t (/(t)− /(t)) = R,

ˆ/t ([(t)− [(t)) = y, [ = b, #, i

R8

S.6 T`K2i`BxiBQMb

Ç 1`Hv KQ/2Hb,7B(t) = · tα(R − t)β

Ç T`K2i2`b `2 +?Qb2M #b2/ QM >2bbBM KBMBKBxiBQM TT`Q+?Ç *M bBKTH2 KQ/2H T`QpB/2 `2HB#H2 mM+2`iBMiv 2biBKi2\Ç *M Bi /2H rBi? /i BM+QMbBbi2M+B2b\

Ç LLS.6 TT`Q+?,

7B(t,Zy) = · tα(R − t)βLL(t)

Ç 7mHHv +QMM2+i2/ JGS Uk@8@j@RVÇ irQ bB;KQB/ ?B//2M Hv2`b M/ HBM2` QmiTmi Hv2`Ç t3 BM/2T2M/2Mi S.6b ⇒ kNe 7`22 T`K2i2`b

Re

*Qbi 7mM+iBQM

Ç q2 KBMBKBx2 i?2 +Qbi 7mM+iBQM,

χk =∑

BD(.B − PB)σ

−RB,D (.D − PD)

Ç .B Bb i?2 2tT2`BK2MiH K2bm`2K2Mi 7Q` TQBMi BÇ PB i?2 i?2Q`2iB+H T`2/B+iBQM 7Q` TQBMi B U= σ ⊗ 7VÇ σBD Bb i?2 +Qp`BM+2 Ki`Bt #2ir22M TQBMib B M/ D rBi? +Q``2+iBQMb

7Q` MQ`KHBxiBQM mM+2`iBMiB2bÇ bmTTH2K2Mi2/ #v //BiBQMH T2MHiv i2`Kb 7Q` TQbBiBpBiv Q#b2`p#H2b

Rd

S`QT;iBM; 2tT2`BK2MiH mM+2ìBMiB2b

:2M2ì2 ìB+BH JQMi2 *`HQ /i `2THB+b 7`QK 2tT2`BK2MiH /iXq2 T2`7Q`K L`2T PURyyyV ib- bKTHBM; Tb2m/Q/i `2THB+b,

.(`)B → .(`)

B + +?QH(Σ)B,DN (y, R), B, D = R..L/i, ` = R...L`2T

q2 Q#iBM L`2T S.6 `2THB+bX LQ bbmKTiBQMb i HH #Qmi i?2:mbbBMBiv Q7 i?2 2``Q`bX

q2 T2`7Q`K +QKT`2bbBQM i2+?MB[m2b 7Q` S.6 /2HBp2`v,

Ç *J*@S.6b, +QKT`2bbBQM H;Q`Bi?K 7Q` J* S.6bXÇ K+k?2bbBM, J* iQ ?2bbBM +QMp2`bBQM iQQH 7Q` S.6bXÇ aJS.6, aT2+BHBx2/ JBMBKH S.6bX

S.6 `2H2b2b `2/m+2 Ryyy `2THB+b iQ RyyX

R3

S`QT;iBM; 2tT2`BK2MiH mM+2ìBMiB2b

:2M2ì2 ìB+BH JQMi2 *`HQ /i `2THB+b 7`QK 2tT2`BK2MiH /iXq2 T2`7Q`K L`2T PURyyyV ib- bKTHBM; Tb2m/Q/i `2THB+b,

.(`)B → .(`)

B + +?QH(Σ)B,DN (y, R), B, D = R..L/i, ` = R...L`2T

q2 Q#iBM L`2T S.6 `2THB+bX LQ bbmKTiBQMb i HH #Qmi i?2:mbbBMBiv Q7 i?2 2``Q`bXq2 T2`7Q`K +QKT`2bbBQM i2+?MB[m2b 7Q` S.6 /2HBp2`v,


S.6 `2H2b2b `2/m+2 Ryyy `2THB+b iQ RyyX

R3

S.6 i 2tKTH2

h?2 T`Q+2/m`2 /2HBp2`b JQMi2 *`HQ `2T`2b2MiiBQM Q7 `2bmHib,

-510 -410 -310 -210 -110

-2

-1

0

1

2

3

4

5

6

7)2xg(x, Q

NNPDF 2.3 NNLO replicas

NNPDF 2.3 NNLO mean value

error bandσNNPDF 2.3 NNLO 1

NNPDF 2.3 NNLO 68% CL band

)2xg(x, Q

h?2 +2Mi`H pHm2 Q7 Q#b2`p#H2b #b2/ QM S.6b `2 Q#iBM2/ rBi?,

⟨O[7]⟩ = RL`2T

L`2T∑

F=RO[7F]

S?2MQK2MQHQ;B+H BKTHB+iBQMb rBHH #2 T`2b2Mi2/ BM i?2 T`HH2H b2+iBQMXRN

PTiBKBxiBQM H;Q`Bi?K

h?2 +m``2Mi TT`Q+? Bb ;2M2iB+ QTiBKBxiBQM- #b2/ QM MQ/H KmiiBQMT`Q##BHBiB2b M/ KQ`2 `2+2MiHv i?2 +Qp`BM+2 Ki`Bt 2pQHmiBQM biì2;v

r → r + η`δ

L`Bi2Bi2

, η = R8, `δ ∼ l(−R, R), `Bi2 ∼ l(R, y)

i 2+? Bi2ìBQM- ;2M2ì2 3y KmiMib M/ b2H2+i #2bi KmiMiX/pMi;2b

Ç aBKTH2 iQ BKTH2K2Mi M/ mM/2`biM/XÇ :QQ/ /2HBM; rBi? +QKTH2t MHviB+ #2?pBQ`XÇ .Q2bMǶi `2[mB`2 2pHmiBM; i?2 ;`/B2MiX

.Bb/pMi;2b

Ç Jv MQi #2 +HQb2 iQ ;HQ#H KBMBKmKXÇ _2[mB`2b KMv 7mM+iBQMb 2pHmiBQMbXÇ L22/b imMBM;X

ky

aiQTTBM;

q2 ?p2 +`Qbb@pHB/iBQM BKTH2K2Mi2/,

Ç q2 bTHBi /i BM i`BMBM; M/ pHB/iBQM b2iXÇ h`BMBM; 7`+iBQM Bb 8yW- /Bz2`2Mi 7Q` 2+? `2THB+XÇ q2 T2`7Q`K i?2 : QM i?2 i`BMBM; b2i 7Q` t2/ MmK#2` Q7

Bi2`iBQMb PUjyyyyVXÇ aiQT i i?2 KBMBKmK Q7 i?2 pHB/iBQM b2i- biQ`BM; i?2 T`K2i2`b

7`QK i?2 `2THB+ i i?i Bi2`iBQMX

kR

oHB/iBQM rBi? +HQbm`2 i2bi

*HQbm`2 i2bib

Ç bbmK2 i?i i?2 mM/2`HvBM; S.6 Bb FMQrM- ;2M2`i2 /i- ~m+imiBQMb`QmM/ i?2 T`2/B+iBQM Q7 i?2 i`m2 S.6X

Ç S2`7Q`K i M/ +QKT`2 iQ mM/2`HvBM; S.6XÇ *?2+F i?i i?2 `2bmHib `2 +QMbBbi2MiX

G2p2H y, 6Bi T`2/B+iBQMb Q7 i?2 i`m2 S.6 rBi?Qmi ~m+imiBQMbX χk/L/i → yX

G2p2H k, :2M2`i2 Tb2m/Q/i `2THB+b QM iQT Q7 `2THB+bX χk/L/i → RXkk

amKK`v

amKK`v

AM bmKK`v,

Ç JG iQ >1S@h> Bb b?QrBM; BMi2`2biBM; TTHB+iBQM QTTQìmMBiB2bXÇ _2bmHib 7`QK i?2 S.6 +QKKmMBiv `2 2M+Qm`;BM;X

Ç lM#Bb2/ TìQM bi`m+im`2 /2i2`KBMiBQMb

6mim`2 /2p2HQTK2Mib,

Ç 6BM/ #2ii2` KQ/2Hb 7Q` S.6 iiBM;Ç AKTH2K2Mi ;`/B2Mi #b2/ QTiBKBx2`bÇ h`v ;2M2ìBp2 KQ/2Hb

kj

h?MFb 7Q` vQm` ii2MiBQM5

kj

S.69G>* iQQHb 7Q` G>* _mM AAU"mii2`rQ`i? 2i HX- `sBp,R8RyXyj3e8V

S.6 `2H2b2b M/ M2r iQQHb

S.6 iQQHb 7Q` i?2 S.69G>*R8,


k9

AMi`Q/m+iBQM

*?HH2M;2.2i2`KBM2 i?2 #2bi +QK#BM2/ S.6 mM+2`iBMiv 7`QK BM/BpB/mH S.6 b2ibX

6`QK kyRy- i?2 S.69G>* q: `2H2b2/ `2+QKK2M/iBQMb- mT/i2/ b2p2`HiBK2b iQ BM+Hm/2 M2r2` p2`bBQMb M/ #m; t2bX

hQr`/b i?2 S.69G>*R8 `2+QKK2M/iBQMAM kyR9fkyR8 JJ>h- *h M/ LLS.6 BKT`Qp2 bB;MB+MiHv ;`22K2Mi /m2iQ M2r /i- #2ii2` i?2Q`v i`2iK2Mi M/ #2ii2` mM/2`biM/BM; Q7 iiBM; Bbbm2bX

k8

AMi`Q/m+iBQM

*?HH2M;2.2i2`KBM2 i?2 #2bi +QK#BM2/ S.6 mM+2`iBMiv 7`QK BM/BpB/mH S.6 b2ibX

6`QK kyRy- i?2 S.69G>* q: `2H2b2/ `2+QKK2M/iBQMb- mT/i2/ b2p2`HiBK2b iQ BM+Hm/2 M2r2` p2`bBQMb M/ #m; t2bX

hQr`/b i?2 S.69G>*R8 `2+QKK2M/iBQMAM kyR9fkyR8 JJ>h- *h M/ LLS.6 BKT`Qp2 bB;MB+MiHv ;`22K2Mi /m2iQ M2r /i- #2ii2` i?2Q`v i`2iK2Mi M/ #2ii2` mM/2`biM/BM; Q7 iiBM; Bbbm2bX

)Z

(MSα0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.12

Cro

ss S

ectio

n (p

b)

28

28.5

29

29.5

30

30.5

31

31.5

Gluon-Fusion Higgs production, LHC 13 TeV

MMHT14CT14NNPDF3.0ABM12HERAPDF2.0JR14VF


( GeV )XM10 210 310

Glu

on -

Glu

on L

umin

osity

0.80.850.9

0.951

1.051.1

1.151.2

1.251.3

CT14NNPDF3.0MMHT14

)=0.118Z

(MSαLHC 13 TeV, NNLO,

k8

S.69G>* `2+QKK2M/iBQMb

S.69G>*RR `2+QKK2M/iBQM

RX lb2 Jahq- *h M/ LLS.6 S.6bkX hF2 i?2 2Mp2HQT2 Q7 mM+2ìBMiB2b b

mM+2ìBMivÇ ;`22K2Mi rb MQi bQ ;QQ/- 2X;X ;;>

+`Qbb b2+iBQM mM+2ìBMiv rb =kt i?2;Bp2M #v Mv BM/BpB/mH b2iX

Ç Qp2`@+QMb2`piBp2, MQ T`QT2` biiBbiB+HK2MBM;

S.69G>*R8 TQbbB#BHBiv

Ç S`QpB/2 +H2` biiBbiB+H BMi2`T`2iiBQMÇ .2HBp2` J* >2bbBM `2T`2b2MiiBQMb

σ29.5 30 30.5 31 31.5 32 32.5 33 33.5

PDF4LHC15_prior

PDF4LHC15_mc

PDF4LHC15_100

PDF4LHC15_30

MMHT14

CT14

NNPDF3.0

MSTW08

CT10

NNPDF2.3


PDF4LHC11 envelopes

ke

S.69G>* `2+QKK2M/iBQMb

S.69G>*RR `2+QKK2M/iBQM

RX lb2 Jahq- *h M/ LLS.6 S.6bkX hF2 i?2 2Mp2HQT2 Q7 mM+2ìBMiB2b b

mM+2ìBMivÇ ;`22K2Mi rb MQi bQ ;QQ/- 2X;X ;;>

+`Qbb b2+iBQM mM+2ìBMiv rb =kt i?2;Bp2M #v Mv BM/BpB/mH b2iX

Ç Qp2`@+QMb2`piBp2, MQ T`QT2` biiBbiB+HK2MBM;

S.69G>*R8 TQbbB#BHBiv

Ç S`QpB/2 +H2` biiBbiB+H BMi2`T`2iiBQMÇ .2HBp2` J* >2bbBM `2T`2b2MiiBQMb σ

29.5 30 30.5 31 31.5 32 32.5 33 33.5

PDF4LHC15_prior

PDF4LHC15_mc

PDF4LHC15_100

PDF4LHC15_30

MMHT14

CT14

NNPDF3.0

MSTW08

CT10

NNPDF2.3


PDF4LHC15 combination

ke

h?2 S.69G>*R8 bi`i2;v

h?2 M2r S.69G>*R8 T`2b+`BTiBQM

RX *QMbi`m+i JQMi2 *`HQ +QK#BM2/ b2i 7`QK ;HQ#H S.6 /2i2`KBMiBQMb

Ç b2ib 2Mi2`BM; BMiQ i?2 +QK#BMiBQM Kmbi biBb7v `2[mB`2K2Mib- 2X;X,;HQ#H /ib2ib- mb2 i?2 :J@o6La- αb b2i iQ i?2 S.: p2`;2X

kX _2/m+2 `2/mM/Mi BM7Q`KiBQMjX .2HBp2` bBM;H2 +QK#BM2/ S.6 b2i @ 2Bi?2` JQMi2 *`HQ Q` >2bbBM 7Q`KX

Monte Carlo Combination

CMC-PDFs MC2H SM-PDFsMeta-PDFs

PDF4LHC15

MC reduction Hessian reductions

Reduction tailoredto specific processes

kd

JQMi2 *`HQ +QK#BMiBQM

h?2 +QK#BMiBQM bi`i2;vq2 b2H2+i i?2 S.6 b2ib i?i 2Mi2` i?2 +QK#BMiBQM⇒ Kmbi #2 `2bQM#Hv +QMbBbi2Mi KQM; i?2KX

:HQ#H b2ib,

( GeV )XM10 210 310

Qua

rk -

Qua

rk L

umin

osity

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25CT14NNPDF3.0MMHT14

)=0.118Z


Ç a2ib `2 +QKTiB#H2XÇ :QQ/ +M/B/i2 7Q` +QK#BMiBQMX

LQM ;HQ#H Y ;HQ#H b2ib,

( GeV )XM10 210 310Q

uark

- Q

uark

Lum

inos

ity0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25CT14

HERAPDF2.0 (EIG+VAR)

)=0.113)Z

(msα=5, f

ABM12 (N

)=0.118Z


Ç *H2` BM+QKTiB#BHBivXÇ GBiiH2 /i- /Bz2`2Mi 2pQHmiBQM-

+?`+i2`BxiBQM Q7 mM+2`iBMivX

k3

h?2 S.69G>*R8 BKTH2K2MiiBQM

h?2 +QK#BM2/ b2ib `2 #b2/ QM biiBbiB+H +QK#BMiBQM Q7,

S.69G>*R8nT`BQ`, *hR9- JJ>hkyR9 M/ LLS.6jXy UJ* b2i- L`2T = NyyV

_2/m+2/ b2ib,

S.69G>*R8n nK+, +QKT`2bb2/ JQMi2 *`HQ b2i rBi? L`2T = RyyXU*J*@S.6b TT`Q+?- `sBp,R8y9Xye9eNV

S.69G>*R8n nRyy, bvKK2i`B+ >2bbBM b2i rBi? L2B; = RyyXUJ*k> TT`Q+?- `sBp,R8y8XyedjeV

S.69G>*R8n njy, bvKK2i`B+ >2bbBM b2i rBi? L2B; = [email protected] TT`Q+?- `sBp,R9y9XyyRjV

JQMi2 *`HQ, +QMiBMb MQM@:mbbBM 72im`2b BKTQ`iMi 7Q` b2`+?2b i ?B;?Kbb2b U?B;? tVX

>2bbBM, mb27mH 7Q` KMv 2tT2`BK2MiH M22/b M/ r?2M mbBM; MmBbM+2T`K2i2`bX Ryy 2B;2Mp2+iQ`b r?2M QTiBKH T`2+BbBQM Bb M22/2/X

kN

*J*@S.6bUa* 2i HX- `sBp,R8y9Xye98NV

h?2 +QKT`2bbBQM B/2

*QKT`2bbBQM B/2,_2/m+2 i?2 bBx2 Q7 S.6 b2i Q7 JQMi2 *`HQ `2THB+brBi? MQfKBMBKH HQbb Q7 BM7Q`KiBQM- 2X;X,

x-510 -410 -310 -210 -110 1

-2

-1

0

1

2

3

4

5

6

7

xg(x,Q), 1000 MC replicas

Members

Central value

Std. deviation

Q = 1.41e+00 GeV

Ge

ne

rate

d w

ith

AP

FE

L 3

.0.0

We

b

x-510 -410 -310 -210 -110 1

-2

-1

0

1

2

3

4

5

6

7

xg(x,Q), 50 compressed replicas

Members

Central value

Std. deviation

Q = 1.41e+00 GeV

Ge

ne

rate

d w

ith

AP

FE

L 3

.0.0

We

b

S`Q#H2K, S`2b2`p2 b Km+? b TQbbB#H2 i?2 mM/2`HvBM; biiBbiB+H/Bbi`B#miBQM Q7 i?2 T`BQ` J* S.6 b2iX

Ç pQB/ #Bb BM i?2 2ti`TQHiBQM `2;BQMXÇ +QMb2`p2 T?vbB+H `2[mB`2K2Mib- 2X;X TQbBiBpBiv- +Q``2HiBQMb- 2i+X

jy

h?2 +QKT`2bbBQM bi`i2;v

q2 /2M2 biiBbiB+H 2biBKiQ` 7Q` i?2 J* T`BQ` b2i,

RX KQK2Mib, +2Mi`H pHm2- p`BM+2- bF2rM2bb M/ Fm`iQbBbkX biiBbiB+H /BbiM+2, i?2 EQHKQ;Q`Qp /BbiM+2jX +Q``2HiBQMb, #2ir22M ~pQ`b i KmHiBTH2 t TQBMib

h?2b2 2biBKiQ`b `2 i?2K +QKT`2/ iQ bm#b2ib Q7 `2THB+b BMi2`+iBp2Hv /`Bp2M#v M 2``Q` 7mM+iBQM- BX2X,

1_6 =∑

F

RLF

∑

B

(*(F)

B − P(F)B

P(F)B

)k

r?2`2 F `mMb Qp2` i?2 MmK#2` Q7 biiBbiB+H 2biBKiQ`b M/

Ç LF Bb MQ`KHBxiBQM 7+iQ` 2ti`+i2/ 7`QK `M/QK `2HBxiBQMbÇ P(F)

B Bb i?2 pHm2 Q7 i?2 2biBKiQ` 7Q` i?2 T`BQ`Ç *(F)

B Bb i?2 +Q``2bTQM/BM; pHm2 7Q` i?2 +QKT`2bb2/ b2i

jR

*J*@S.6b UF S.69G>*R8n nK+V

:QQ/ ;`22K2Mi #2ir22M i?2 S.69G>*R8nT`BQ` M/ *J*@S.6b 7`QK MmK#2` Q7 +QKT`2bb2/ `2THB+b L`2T > 8y- 2X;X,

x 5−10 4−10 3−10 2−10 1−10

) [re

f] 2

) / g

( x,

Q2

g ( x

, Q

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

PDF4LHC15_nnlo_prior

PDF4LHC15_nnlo_mc

)=0.118Z

(MSα, 2=100 GeV2NNLO, Q

x 5−10 4−10 3−10 2−10 1−10

) [re

f] 2

( x,

Qd

) /

2 (

x, Q

d

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

PDF4LHC15_nnlo_prior

PDF4LHC15_nnlo_mc

)=0.118Z

(MSα, 2=100 GeV2NNLO, Q

( GeV )XM10 210 310

Glu

on -

Glu

on L

umin

osity

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 PDF4LHC_nnlo_prior

PDF4LHC_nnlo_mc

)=0.118Z


( GeV )XM10 210 310

Qua

rk -

Antiq

uark

Lum

inos

ity

0.85

0.9

0.95

1

1.05

1.1

1.15


PDF4LHC_nnlo_mc

)=0.118Z


jk

*J*@S.6b UF S.69G>*R8n nK+V

_2bQM#H2 ;`22K2Mi b r2HH 7Q` i?2 +Q``2HiBQMb #2ir22M /Bz2`2Mi S.6~pQm`b M/ BM+HmbBp2 +`Qbb@b2+iBQMbX

bBKBH` MmK#2` Q7 `2THB+b 7`QK 2+? Q7 i?2 i?`22 b2ib Bb miQKiB+HHvb2H2+i2/ #v i?2 +QKT`2bbBQM H;Q`Bi?K⇒LLS.6jXy, kj `2THB+bc *hR9, je `2THB+b- JJ>hR9, jk `2THB+b

jj

J*k> J2i S.6bUa* 2i HX- `sBp,R8y9XyedjeV

>2bbBM `2T`2b2MiiBQMb

S`Q#H2K //`2bb2/ ?2`2,.2i2`KBM2 M mM#Bb2/ >2bbBM `2T`2b2MiiBQM 7Q` J* S.6bX

J*k> ai`i2;v,mb2 J* `2THB+b i?2Kb2Hp2b b i?2#bBb Q7 i?2 HBM2` `2T`2b2MiiBQMXmb2 S`BM+BTH *QKTQM2Mi MHvbBbUS*V iQ `2T`Q/m+2 S.6 +Qp`BM+2Ki`Bt rBi? `#Bi``v T`2+BbBQMX

[email protected] ai`i2;v,2+? J* `2THB+ Bb `2@ii2/ mbBM; ~2tB#H2 ǳK2i@T`K2i`BxiBQMǴX

i?2 #2bi +QMbi`BM2/ +QK#BMiBQM`2 7QmM/ #v /B;QMHBxiBQM Q7 i?2+Qp`BM+2 Ki`Bt QM i?2 S.6 bT+2

j9

J*k> UF S.69G>*R8n nRyyV

>2bbBM `2T`2b2MiiBQM Q7 i?2 S.69G>*R8nT`BQ` ?b #22M +QMbi`m+i2/ mbBM;

Ç J*k> ⇒ S.69G>*R8n nRyy rBi? L2B; = Ryy U?B;? ++m`+vVÇ [email protected] ⇒ S.69G>*R8n njy rBi? L2B; = jy

( GeV )XM10 210 310

Glu

on -

Glu

on L

umin

osity

0.85

0.9

0.95

1

1.05

1.1

1.15


PDF4LHC_nnlo_100

PDF4LHC_nnlo_30

)=0.118Z


( GeV )XM10 210 310

Qua

rk -

Antiq

uark

Lum

inos

ity0.85

0.9

0.95

1

1.05

1.1

1.15


PDF4LHC_nnlo_100

PDF4LHC_nnlo_30

)=0.118Z


1t+2HH2Mi H2p2H Q7 ;`22K2Mi 7Q` S.6b M/ HmKBMQbBiB2b b +QKT`2/ rBi? i?2T`BQ`X

j8

J*k> UF S.69G>*R8n nRyyV +Q``2HiBQMb

1t+2HH2Mi ;`22K2Mi rBi? i?2 T`BQ` 7Q` S.6 +Q``2HiBQM M/ Q#b2`p#H2+Q``2HiBQMX

hBMv `2bB/mH /Bz2`2M+2b i i?2 H2p2H Q7 72r T2`+2Mi- B``2H2pMi 7Q` G>*T?2MQK2MQHQ;vX

je

:mbbBMBiv Q7 i?2 S.69G>*R8 +QK#BMiBQMb

J*k> rQ`Fb #2bi 7Q` :mbbBM #BMb M/ r?2M mbBM; i?2 `2bmHib b :mbbBMX

*J* rQ`Fb #2bi 7Q` MQM@:mbbBM #BMb- r?2M i`2iBM; i?2 `2bmHib b J*X

jd

aJS.6Ua* 2i HX- `sBp,ReyRXyyyy8V

aT2+BHBx2/ JBMBKH S.6b

A/2 Qp2`pB2r1+B2Mi M/ ++m`i2 S.6 T`Q+2bb@bT2+B+ S* >2bbBM `2/m+iBQM H;Q`Bi?KX

Ç S`BQ` S.6- HBbi Q7 Q#b2`p#H2b =⇒ _2/m+2/ `2T`2b2MiiBQM UaJS.6V

The SM-PDFs strategy

Custom observables

Theoreticalpredictions

Kinematic sampling

SM-PDFs

Input PDF set

APPLgrid

Plain files

LHAPDF

LHAPDF gridReached

Tolerance?

SM-PDF algorithm:

Eigenvector collection

Compute uncertaintiesValidation plots

Data output

NoYes

Check non-linear effects

Next observable

Orthogonal projection

j3

1tKTH2 +b2b

q2 ?p2 ;2M2`i2/ aJS.6b 7Q` i?2 KQbi BKTQ`iMi >B;;b T`Q/X T`Q+2bb2b,

S`Q+2bb S.69G>*R8nT`BQ` LLS.6jXy JJ>hR9h_ = 8% h_ = Ry% h_ = 8% h_ = Ry% h_ = 8% h_ = Ry%

;; → ? 9 8 9 9 j jo"6 ?DD d 8 Ry 8 9 j

?q e 8 e 9 e j?w RR d e 9 3 8?ii j k 9 9 j k

hQiH ? R8 RR Rj 3 3 d

M/ i?2 KBM #+F;`QmM/b,

S`Q+2bb S.69G>*R8nT`BQ` LLS.6jXy JJ>hR9h_ = 8% h_ = Ry% h_ = 8% h_ = Ry% h_ = 8% h_ = Ry%

? R8 RR Rj 3 3 dii 9 9 8 9 j j

q, w R9 RR Rj 3 Ry NG//2` Rd R9 R3 RR Ry Ry

h_ Ub2i #v mb2`V Bb i?2 KtBKmK HHQr2/ /2pBiBQM 7`QK i?2 T`BQ` 7Q` Mv #BMX⇒ hvTB+H /Bz2`2M+2 Bb Km+? bKHH2`X

jN

G//2` aJS.6

JmHiBTH2 T`Q+2bb2b +M #2 2+B2MiHv bi+F2/ iQ;2i?2`X

9y

aJS.6 bi#BHBiv

EBM2KiB+H `M;2b i?i /Qm#H2 i?Qb2 mb2/ b BMTmi UT?h M/ v?V

"`2F/QrM QMHv r?2M ;QBM; BM 2ti`2K2 `2;BQMb UH`;2 |η|V,

9R

h?2 S.69G>*R8 /2HBp2`#H2b

G>S.6e ;`B/ SiQX 1``Q`hvT2 LK2K αa(Kkw)

S.69G>*R8nUMVMHQnK+ ULVLGP `2THB+b Ryy yXRR3S.69G>*R8nUMVMHQnRyy ULVLGP bvKK?2bbBM Ryy yXRR3S.69G>*R8nUMVMHQnjy ULVLGP bvKK?2bbBM jy yXRR3S.69G>*R8nUMVMHQnK+nT/7b ULVLGP `2THB+bYb Ryk K2K y,Ryy→yXRR3

K2K RyR→yXRRe8K2K Ryk→yXRRN8

S.69G>*R8nUMVMHQnRyynT/7b ULVLGP bvKK?2bbBMYb Ryk K2K y,Ryy→yXRR3K2K RyR→yXRRe8K2K Ryk→yXRRN8

S.69G>*R8nUMVMHQnjynT/7b ULVLGP bvKK?2bbBMYb jk K2K y,jy→yXRR3K2K jR→yXRRe8K2K jk→yXRRN8

S.69G>*R8nUMVMHQnbp` ULVLGP @ R K2K y→yXRRe8K2K R→yXRRN8

h#H2 R, amKK`v Q7 i?2 +QK#BM2/ S.69G>*R8 b2ib rBi? MKt7 = 8X

9k

h?2 S.69G>*R8 /2HBp2`#H2b

G>S.6e ;`B/ SiQX 1``Q`hvT2 LK2K α(M7=8)a (Kk

w)

S.69G>*R8nMHQnM79nRyy LGP bvKK?2bbBM Ryy yXRR3S.69G>*R8nMHQnM79njy LGP bvKK?2bbBM jy yXRR3S.69G>*R8nMHQnM79nRyynT/7b LGP bvKK?2bbBMYb Ryk K2K y,Ryy→yXRR3

K2K RyR→yXRRe8K2K Ryk→yXRRN8

S.69G>*R8nMHQnM79njynT/7b LGP bvKK?2bbBMYb jk K2K y,jy→yXRR3K2K jR→yXRRe8K2K jk→yXRRN8

S.69G>*R8nMHQnM79nbp` LGP @ R K2K y→yXRRe8K2K R→yXRRN8

h#H2 k, amKK`v Q7 i?2 +QK#BM2/ S.69G>*R8 BM i?2 M7 = 9X

9j


h?2 +QK#BMiBQM bi`i2;v

RX q2 b2H2+i i?2 S.6 b2ib i?i 2Mi2` i?2 +QK#BMiBQM⇒ Kmbi #2 `2bQM#Hv +QMbBbi2Mi KQM; i?2KX

kX h`Mb7Q`K i?2 >2bbBM S.6 b2ib BMiQ i?2B` JQMi2 *`HQ `2T`2b2MiiBQMUqii M/ h?Q`M2 ǶRkV,

6F = 6([y) +Rk

L2B;∑

D=R

[6([+

D )− 6([−D )]

_FD , F = R, . . . ,L`2T

jX *QK#BM2 i?2 bK2 MmK#2` Q7 `2THB+b 7`QK 2+? Q7 i?2 T`BQ` b2ib-bbmKBM; 2[mH r2B;?i BM i?2 +QK#BMiBQM UBX2X M mMr2B;?i2/ b2iVX

S.69G>*R8, r2 +QK#BM2 L`2T = jyy `2THB+b 7`QK LLS.6jXy- *hR9 M/JJ>hkyR9- ?Qr2p2` Mv Qi?2` +?QB+2 Bb TQbbB#H2X

99


h?2 `2bmHiBM; +QK#BM2/ JQMi2 *`HQ b2i ?b biiBbiB+H T`QT2`iB2b r?B+? H2/iQ bKHH2` mM+2`iBMiB2b i?M i?2 S.69G>*RR 2Mp2HQT2X

Cro

ss-S

ect

ion

(pb

)

41

41.5

42

42.5

43

43.5

44

44.5

45

=0.118sαggH, ggHiggs NNLO, LHC 13 TeV,

NNPDF3.0

MMHT14

CT14

MC900

Envelope

=0.118sαggH, ggHiggs NNLO, LHC 13 TeV,

Cro

ss-S

ect

ion

(nb

)6.8

7

7.2

7.4

7.6

7.8

=0.118sα, VRAP NNLO, LHC 13 TeV, +W

NNPDF3.0

MMHT14

CT14

MC900

Envelope

=0.118sα, VRAP NNLO, LHC 13 TeV, +W

S`QT2` i`2iK2Mi Q7 QmiHB2`b ⇒ i?2 2Mp2HQT2 ;Bp2b KQ`2 r2B;?i iQ QmiHB2`bX

98


S.69G>*R8nT`BQ` Nyy J* `2THB+b `2[mB`2/ iQ bi#BHBx2 i?2 +QK#BMiBQMX

x

5−

10 4−103−

10 2−10 1−10

(x,Q

) (r

ef)

g(x

,Q)

/ g

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

=0.118, Q = 100 GeVS

αNNLO,

MC900

NNPDF3.0

CT14

MMHT14

=0.118, Q = 100 GeVS

αNNLO,

x

5−

10 4−103−

10 2−10 1−10

(x,Q

) (r

ef)

u(x

,Q)

/ u

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

=0.118, Q = 100 GeVS

αNNLO,

MC900

NNPDF3.0

CT14

MMHT14

=0.118, Q = 100 GeVS

αNNLO,

Abbm2b #27Q`2 i?2 /2p2HQTK2Mi Q7 `2/m+iBQM bi`i2;B2b,

Ç iQQ KMv `2THB+b 7Q` T`+iB+H TTHB+iBQMb UL`2T = NyyVÇ MQ TQbbB#H2 >2bbBM `2T`2b2MiiBQMÇ MQ `2/m+2/ rv iQ T`2b2`p2 MQM@:mbbBM 72im`2b

9e


h?2 J* +QK#BMiBQM Bb mbmHHv :mbbBM #mi BM KMv +b2b MQM@:mbbBM72im`2b `2 Q#b2`p2/X

S`iB+mH` BKTQ`iMi 7Q` "aJ b2`+?2b- r?B+? `2Hv QM S.6b BM `2;BQMb r?2`2S.6 2``Q`b `2 H`;2X

:mbbBM

x*PDF( x ,Q )

7.4 7.6 7.8 8 8.2 8.4

Pro

babili

ty p

er

bin

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

g( x=0.01, Q=100 GeV )

NNPDF3.0

CT14

MMHT14

MC900

Gaussian

g( x=0.01, Q=100 GeV )

LQM@:mbbBM

x*PDF( x ,Q )

0.08 0.1 0.12 0.14 0.16 0.18 0.2

Pro

babili

ty p

er

bin

0

0.1

0.2

0.3

0.4

0.5

0.6

s( x=0.05, Q=100 GeV )

NNPDF3.0

CT14

MMHT14

MC900

Gaussian

s( x=0.05, Q=100 GeV )

9d


h?2 H;Q`Bi?K b2H2+ib `2THB+b 7`QK i?2 T`BQ` i?i KBMBKBx2 i?2 2``Q` 7mM+iBQMXh?2 KBMBKBxiBQM Bb /`Bp2M #v ;2M2iB+ H;Q`Bi?KX

oHB/iBQM, 2biBKiQ`b- S.6 THQib- i?2Q`2iB+H T`2/B+iBQMb- /BbiM+2b- χk iQ2tT2`BK2MiH /i- 2i+X

93


h2bi +b2,1tKTH2 Q7 1_6 KBMBKBxiBQM 7Q` L`2T = Ryyy 7`QK LLS.6jXy LGPX

Iteration0 2000 4000 6000 8000 10000 12000 14000

TO

TE

RF

1

Total error function minimization - 1000 replicas prior

10 compressed replicas









+QKT`2bbQ` pRXyXy: S`K2i2`b

LKt;2M R8yyy

LKmi 8Lt dy

tKBM Ry−8

tKt y.NM7 dZy mb2`@/2M2/

L+Q``t 8

L`M/ Ryyy

Ç h?2 H;Q`Bi?K `2+?2b i?2 bi#BHBiv THi2m 7i2` kF Bi2`iBQMbXÇ G`;2 T`BQ` Q7 J* `2THB+b ⇒ BM+`2b2b TQbbB#H2 +QK#BMiBQMbX

9N


JQK2Mi 2biBKiQ`b 7Q` i?2 +QKT`2bbBQM M/ `M/QK b2H2+iBQMbX@ ?Q`BxQMiH HBM2b ⇒ HQr2` e3W +XHX 7Q` `M/QK b2H2+iBQM rBi? L`2T = Ryy

Replicas10

210

1

10

210

310

410

510

610

ERF Central Value

Compressed

Random Mean (1k trials)

Random Median (1k trials)

Random 50% c.l. (1k trials)



Replicas10

210

-110

1

10

210

ERF Standard deviation

Compressed

Random Mean (1k trials)

Random Median (1k trials)




Ç am#biMiBH BKT`Qp2K2Mib b +QKT`2/ iQ `M/QK b2H2+iBQMbXÇ *QKT`2bbBQM Bb #H2 iQ bm++2bb7mHHv `2T`Q/m+2 ?B;?2` KQK2Mib M/

+Q``2HiBQMbXÇ AM i?Bb i2bi +b2 L`2T = 8y `2 2[mBpH2Mi iQ J* ib rBi? Ryy `2THB+bX

8y

G>* S?2MQK2MQHQ;v

*J*@S.6b HbQ pHB/i2/ 7Q` G>* BM+HmbBp2 +`Qbb@b2+iBQMb M/ /Bz2`2MiBH/Bbi`B#miBQMb- BM+Hm/BM; +Q``2HiBQMbX

Ratio to MC9000.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05

=0.118sαLHC 13 TeV, NNLO,

gg->h

+W

-W

0Z

tT

= 100repN

=0.118sαLHC 13 TeV, NNLO,

Ratio to original Monte Carlo combined PDFs0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

=0.118, NLOsαLHC 7 TeV,

Low-Mass DY

High-Mass DY

Forw DY

W+charm

Cent Jets

Forw Jets

= 100repN

=0.118, NLOsαLHC 7 TeV,

Co

rre

latio

n c

oe

ffic

ien

t

1−

0.5−

0

0.5

1

Correlation Coefficient for gg->h

gg->h -W tT

+W 0Z

= 100repN

Reference

Compressed

Random (68% CL)

Correlation Coefficient for gg->h

Co

rre

latio

n c

oe

ffic

ien

t

1−

0.5−

0

0.5

1

Correlation Coefficient for tT

gg->h -W tT

+W 0Z

= 100repN

Reference

Compressed

Random (68% CL)

Correlation Coefficient for tT

8R

LQM@:mbbBM 72im`2b BM G>* +`Qbb@b2+iBQMb

LQM@:mbbBM 72im`2b `2 HbQ +H2`Hv Q#b2`p2/ i i?2 H2p2H Q7 G>* T`Q+2bb2b-2X;X KQbi 7Q`r`/ #BM Q7 i?2 *Ja qY+?`K /Bz2`2MiBH +`Qbb@b2+iBQMK2bm`2K2Mi M/ .u K2bm`2K2Mi 7`QK G>*#X

>2bbBM `2/m+iBQM 7BHb #v +QMbi`m+iBQM r?2M `2T`Q/m+BM; bm+? 72im`2bX

>Qr2p2`- BM `2;BQMb r?2`2 i?2 :mbbBM TT`QtBKiBQM Bb `2bQM#H2- QM2b?QmH/ mb2 >2bbBM `2T`2b2MiiBQMX

8k

:2M2`H bi`i2;v

RX :Bp2M JQMi2 *`HQ T`BQ` b2i Q7 S.6b

7(F)α,K+F=R,...,L`2T , α = ;, m, /, b, . . . ,

kX 6Bt i?2 +2Mi`H pHm2 iQ #2 i?2 bK2 b i?2 T`BQ`,

7(y)α,?2bbBM = 7(y)α,K+

jX q2 /2M2 i?2 Ki`Bt 7Q` i?2 /2pBiBQMb r`i +2Mi`H pHm2,

sHF(Z) ≡ 7(F)α,K+(tB,Z)− 7(y)α (tB,Z), H ≡ Lt(α− R) + B

9X h?2 +Qp`BM+2 Ki`Bt Bb ;Bp2M BM i2`Kb Q7 s,

+QpT/7BD,αβ(Z) ≡ R

L`2T − Rssi

8j

:2M2`H bi`i2;v

ao. /B;QMH `2T`2b2MiiBQM Q7 i?2 +Qp`BM+2 Ki`Bt BM i2`Kb Q7 `2THB+b Bb 7QmM/#v ao. Q7 i?2 Ki`Bt s,

s = laoi,

o Bb M Q`i?Q;QMH L`2T × L`2T Ki`Bt Q7 +Q2+B2Mib- M/

so,

T`QpB/2b `2T`2b2MiiBQM Q7 i?2 KmHiB;mbbBM +Qp`BM+2 Ki`Bt BM i2`Kb Q7 i?2Q`B;BMH `2THB+bX

S* _2/m+iBQMJMv 2B;2Mp2+iQ`b H2/ iQ p2`v bKHH +QMi`B#miBQM iQ i?2 +Qp`BM+2 Ki`Bt⇒ r2 +M b2H2+i bKHH2` b2i Q7 L2B;- rBi? H`;2bi 2B;2MpHm2b- r?B+? biBHHT`QpB/2b ;QQ/ TT`QtBKiBQM iQ i?2 +Qp`BM+2 Ki`BtX

89

:2M2`H bi`i2;v

h?2 S* QTiBKBxiBQM `2iBMb i?2 T`BM+BTH +QKTQM2Mib- BX2X i?2 H`;2bibBM;mH` pHm2bX

Ç l- a `2 `2TH+2/ #v i?2B` bm#Ki`B+2b m- b `2bT2+iBp2HvXÇ /BK m = LtL7 × L2B; M/ /BK b = L2B; × L`2T

Ç PMHv i?2 L`2T × L2B; Q`i?Q;QMH mTT2` H27i bm#Ki`Bt Q7 o +QMi`B#mi2bÇ h?Bb Bb i?2 T`BM+BTH bm#Ki`Bt S Q7 o,

SFB = oFB, F = R, . . .L`2T; B = R, . . . ,L2B;

h?mb r2 r`Bi2 i?2 >2bbBM 2B;2Mp2+iQ`b b HBM2` +QK#BMiBQM Q7 `2THB+b,

7(B)α,?2bbBM(tD,Z) = 7(y)α (tD,Z) + sHFSFB, H ≡ Lt(α− R) + D

= 7(y)α (tD,Z) +

L`2T∑

F=R(B)

F

(7(F)α,K+(tD,Z)− 7(y)α (tD,Z)

)

LQi2 i?2 (B)F BM/2T2M/2M+2 BM (t,Z)X Ai iF2b +`2 Q7 2pQHmiBQM miQKiB+HHvX

88

_Q#mbiM2bb M/ ;mbbBMBiv Q7 i?2 S.69G>*R8 +QK#BMiBQMb

A/2h2bi i?2 ++m`+v M/ :mbbBMBiv Q7 i?2 S.69G>*R8 b2ibX

Ç o2`B7v i?2 `M;2 Q7 pHB/Biv Q7 T`2/B+iBQM mbBM; /i BM+Hm/2/ BM S.6 ibXÇ .Bb+`BKBMi2 :mbbBMBiv Q7 T`2/B+iBQMb ⇒ p2`B7v J* pb >2bbBM

`2T`2b2MiiBQMbX

_2bmHib 2H#Q`i2/ 7Q` i?2 G2b >Qm+?2b kyR8 T`Q+22/BM;bX

8e

_Q#mbiM2bb Q7 i?2 S.69G>*R8 +QK#BMiBQMb

q2 ?p2 +QKTmi2/ T`2/B+iBQMb rBi? S.69G>*R8nT`BQ` M/ i?2 i?`22 `2/m+2/b2ib- 7Q` HH /i ?/`QMB+ /i BM+Hm/2/ BM i?2 LLS.6jXy /ib2iX

.2pBiBQMb `2 ;2M2`HHv bKHH- M/ +QM+2Mi`i2/ BM `2;BQMb BM r?B+?2tT2`BK2MiH BM7Q`KiBQM Bb b+`+2 M/ S.6 mM+2`iBMiB2b `2 H`;2bi ⇒ H`;2 tM/ H`;2 ZX

8d


83


AM Q`/2` iQ 2biBKi2 i?2 ;mbbBMBiv Q7 T`2/B+iBQMb r2 +QMbi`m+i +QMiBMmQmbT`Q##BHBiv /2MbBiv 7`QK JQMi2 *`HQ bKTH2 UE2`M2H .2MbBiv 1biBKi2V,

S(σB) =R

L`2T

L`2T∑

F=RE(σB − σ(F)

B ), B = R, . . . ,L/i

q2 mb2 i?2 EmHH#+F@G2B#H2` /Bp2`;2M+2 iQ K2bm`2 ?Qr Km+? BM7Q`KiBQM r2`2 HQQbBM; #v TT`QtBKiBM; i?2 T`BQ` S(σ) rBi? i?2 /Bbi`B#miBQM bTMM2/ 7Q`K2+? Q7 i?2 QTiBKBx2/ `2T`2b2MiiBQMb Z(σ)X

.(B)EG(S|Z) =

ˆ ∞

−∞

(S(σB) ·

HQ;S(σB)HQ;Z(σB)

)/σB

8N


q2 mb2 i?2 EmHH#+F@G2B#H2` /Bp2`;2M+2 iQ K2bm`2 ?Qr Km+? BM7Q`KiBQM r2`2 HQQbBM; #v TT`QtBKiBM; i?2 T`BQ` S(σ) rBi? i?2 /Bbi`B#miBQM bTMM2/ 7Q`K2+? Q7 i?2 QTiBKBx2/ `2T`2b2MiiBQMb Z(σ)X

.(B)EG(S|Z) =

ˆ ∞

−∞

(S(σB) ·

HQ;S(σB)HQ;Z(σB)

)/σB

q2 +QKT`2 i?2 E.1 Q7 i?2 T`BQ` rBi?

Ç :mbbBM ;Bp2M #v µ = ⟨σB⟩B- σ = RL−R

√∑(σB − µ)kX

Ç h?2 J*k> :mbbBMXÇ h?2 *J* E.1X

>2`2 r2 ?p2 mb2/ i?2 aJS.6 /ib2iX

ey


SQBMib BM i?2 /B;QMH ⇒ ;`22b 2t+iHv rBi? i?2 Tm`2Hv :mbbBM TT`QtX

P`M;2 TQBMib #2HQr /B;QMH ⇒ *J* #2ii2` i?M J*k>

eR


EG /Bp2`;2M+2 T`Q+2bb #v T`Q+2bb,

ek


q2 ?p2 +QKTmi2/ T`2/B+iBQMb rBi? S.69G>*R8nT`BQ` M/ i?2 i?`22 `2/m+2/b2ib- 7Q` HH /i BM i?2 LLS.6jXy /ib2iX

x-6

10-5

10 -410-3

10 -210 -110 1

]2

[ G

eV

2 T / p

2 / M

2Q

1

10

210

310

410

510

610

710FT DIS

HERA1

FT DY

TEV EW

TEV JET

ATLAS EW

LHCB EW

LHC JETS

HERA2

ATLAS JETS 2.76TEV

ATLAS HIGH MASS

ATLAS WpT

CMS W ASY

CMS JETS

CMS WC TOT

CMS WC RAT

LHCB Z

TTBAR

NNPDF3.0 NLO dataset

ej

aJS.6 #+FmT @ H;Q`Bi?K bi`i2;v

6QHHQrBM; i?2 J*k> S* K2i?Q/QHQ;v r2 +M M/ bm#bT+2 rBi? bKHH2`MmK#2` Q7 T`K2i2`b r?B+? QTiBKBx2b i?2 ;`22K2Mi 7Q` bQK2 [mMiBiB2bX

s = sS ∈ RLtLT/7 × RL2B;≪L`2T

q2 +M ;`2iHv BKT`Qp2 i?2 `2/m+iBQM #v i`;2iBM; bT2+B+ T`Q+2bb2b,

σB, B = R, . . . ,Lσ

bσB =

⎛

⎝ RL`2T − R

L`2T∑

F=R

(σ(F)

B − σ(y)B

)k⎞

⎠

Rk

h?2 rQ`bi@+b2 ++m`+v i`;2i +M #2 imM2/ #v mb2`,

h_ < KtB∈(R,Lσ)

∣∣∣∣R−bσB

bσB

∣∣∣∣

h?Bb Bb BKTH2K2Mi2/ BM M BMi2`+iBp2 T`Q+2/m`2X

e9

aJS.6 #+FmT @ b2H2+iBQM H;Q`Bi?K

6Q` 2+? Bi2`iBQM- b2H2+i TQBMib BM (t,α,Z) +Q``2Hi2/ rBi? p`BiBQMb BM σ

ρ(tB,Z,α,σ) =L`2T

L`2T − R⟨s(Zα)HF · (σ(F) − σ(y))⟩`2T − ⟨s(Zσ)HF⟩`2T · ⟨σ(F) − σ(y)⟩`2T

bS.6α · bσ

Ξ = (sB,α) : ρ(sB,Zα,α,σ) ≥ i · ρKt, s → sΞ(Zσ)

h?2 +Q``2HiBQM i?`2b?QH/ i Bb i?2 QMHv 7`22 T`K2i2` Q7 i?2 H;Q`Bi?K ⇒i = y.N QTiBKH +?QB+2X

e8

aJS.6 #+FmT @ Qì?Q;QMH T`QD2+iBQM H;Q`Bi?K

h?Bb TT`Q+? HHQrb iQ 2+B2MiHv ;2M2`HBx2 iQ T`Q+2bb2b rBi? bBKBH` S.6/2T2M/2M+2- KFBM; i?2 H;Q`Bi?K bi#H2X

q2 +QKTmi2 i?2 ao. Q7 sΞ M/ b2H2+i QM2 2B;2Mp2+iQ`,

sΞ(Zα) = laoi

(S · _) = o ∈ RL`2T ×(RRRL`2T−R

)

q2 T`QD2+i Qmi i?2 b2H2+i2/ 2B;2Mp2+iQ` 7Q` i?2 M2ti Bi2ìBQM

s → s_

q2 Bi2ì2 Ub2H2+i KQ`2 2B;2Mp2+iQ`bV mMiBH r2 K22i i?2 iQH2`M+2 +`Bi2`B 7Qì?2 +m``2Mi Q#b2`p#H2- M/ KQp2 iQ i?2 M2ti Q#b2`p#H2- mMiBH r2 `2T`Q/m+2 HHX

ee

aJS.6 #+FmT @ SSG;`B/b

AMTmi +`Qbb@b2+iBQMb 7Q` [email protected] 7Q` >B;;b T?vbB+bT`Q+2bb /Bbi`B#miBQM ;`B/ MK2 L#BMb `M;2 FBMX +mib;; → ? BM+H tb2+ ;;?nRji2p R @ @

/σ//T?i ;;?nTinRji2p Ry (y-kyy) :2o @

/σ//v? ;;?nvnRji2p Ry (@kX8-kX8) @o"6 ?DD BM+H tb2+ p#7?nRji2p R @ @

/σ//T?i p#7?nTinRji2p 8 (y-kyy) :2o @

/σ//v? p#7?nvnRji2p 8 (@kX8-kX8) @?q BM+H tb2+ ?rnRji2p R @ Th(H) ≥ Ry :2o- |ηH| ≤ k.8

/σ//T?i ?rnTinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8

/σ//v? ?rnvnRji2p Ry (@kX8-kX8) Th(H) ≥ Ry :2o- |ηH| ≤ k.8?w BM+H tb2+ ?xnRji2p R @ Th(H) ≥ Ry :2o- |ηH| ≤ k.8

/σ//T?i ?xnTinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8

/σ//v? ?xnvnRji2p Ry (@kX8-kX8) Th(H) ≥ Ry :2o- |ηH| ≤ k.8?ii BM+H tb2+ ?ii#`nRji2p R @ @

/σ//T?i ?ii#`nTinRji2p Ry (y-kyy) :2o @

/σ//v? ?ii#`nvnRji2p Ry (@kX8-kX8) @

ed


AMTmi +`Qbb@b2+iBQMb 7Q` [email protected] 7Q` ii T?vbB+bT`Q+2bb /Bbi`B#miBQM ;`B/ MK2 L#BMb `M;2 FBMX +mib

ii BM+H tb2+ ii#`nRji2p R @ @/σ//Ti

i ii#`ni#`TinRji2p Ry (9y-9yy) :2o @/σ//vi ii#`ni#`vnRji2p Ry (@kX8-kX8) @/σ//Ti

i ii#`niTinRji2p Ry (9y-9yy) :2o @/σ//vi ii#`nivnRji2p Ry (@kX8-kX8) @

/σ//Kii ii#`nii#`BMpKbbnRji2p Ry (jyy-Ryyy) @/σ//Tii

i ii#`nii#`TinRji2p Ry (ky-kyy) @/σ//vii ii#`nii#`vnRji2p Rk (@j-j) @

e3


AMTmi +`Qbb@b2+iBQMb 7Q` [email protected] 7Q` 2H2+i`Qr2F #QbQM T`Q/m+iBQM T?vbB+bT`Q+2bb /Bbi`B#miBQM ;`B/ MK2 L#BMb `M;2 FBMX +mib

w BM+H tb2+ xnRji2p R @ Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//TH−

i xnHKTinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//vH− xnHKvnRji2p Ry (@kX8-kX8) Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//TH+

i xnHTTinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//vH− xnHTvnRji2p Ry (@kX8-kX8) Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//Tx

i xnxTinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//vx xnxvnRji2p 8 (@9-9) Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//KHH xnHTHKBMpKbbnRji2p Ry (8y-Rjy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//THH

i xnHTHKTinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8

q BM+H tb2+ rnRji2p R @ Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//φ rn+T?BnRji2p Ry (@R-R) Th(H) ≥ Ry :2o- |ηH| ≤ k.8

/σ//1KBbbi rn2iKBbbnRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8

/σ//THi rnHTinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8

/σ//vH rnHvnRji2p Ry (@kX8-kX8) Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//Ki rnKinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//Tr

i rnrTinRji2p Ry (y-kyy) :2o Th(H) ≥ Ry :2o- |ηH| ≤ k.8/σ//vr rnrvnRji2p Ry (@9-9) Th(H) ≥ Ry :2o- |ηH| ≤ k.8

eN

aJS.6 `2bmHib

dy

_B2KMM@h?2i "QHixKMM K+?BM2#b2/ QM `sBp,RdRkXyd83R M/ `sBp,R3y9Xydde3.X E`2~- aX *``xx- "X >;?B;?i- CX E?H2M

.MB2H E`2~ M/ ai27MQ *``xxhbBM;?m BM JG BM ;2QK2i`v M/ T?vbB+b- RR@R8 CmM2 kyR3- aMv

1m`QT2M P`;MBxiBQM 7Q` Lm+H2` _2b2`+? U*1_LV

+FMQrH2/;2K2Mi, h?Bb T`QD2+i ?b `2+2Bp2/ 7mM/BM; 7`QK >A**lS 1_* *QMbQHB/iQ`;`Mi UeR98ddV M/ #v i?2 1m`QT2M lMBQMǶb >Q`BxQM kyky `2b2`+? M/ BMMQpiBQMT`Q;`KK2 mM/2` ;`Mi ;`22K2Mi MQX d9yyyeX

PDFN 3Machine Learning • PDFs • QCD

https://arxiv.org/abs/1712.07581

https://arxiv.org/abs/1804.07768

AMi`Q/m+iBQM

AMi`Q/m+iBQM

q2 bi`i2/ i?Bb T`QD2+i BKBM; iQ #mBH/ KQ/2H rBi?,

Ç r2HH bmBi2/ 7Q` T/7 2biBKiBQM M/ T/7 bKTHBM;Ç #mBHi@BM T/7 MQ`KHBxiBQM U+HQb2 7Q`K 2tT`2bbBQMVÇ p2`v ~2tB#H2 rBi? bKHH MmK#2` Q7 T`K2i2`b

q2 /2+B/2/ iQ HQQF i 2M2`;v KQ/2Hb- bT2+B+HHv "QHixKMM J+?BM2bX

R

h?2Q`v

"QHixKMM K+?BM2

:`T?B+H `2T`2b2MiiBQM,

(>BMiQM- a2DMQrbFB ǵ3e)

ǳoBbB#H2Ǵ b2+iQ`

ǳ>B//2MǴ b2+iQ`"BM`v pHm2/bii2b y, R

*QMM2+iBQMKi`B+2b

k

"QHixKMM K+?BM2

:`T?B+H `2T`2b2MiiBQM, (>BMiQM- a2DMQrbFB ǵ3e)

ǳoBbB#H2Ǵ b2+iQ`

ǳ>B//2MǴ b2+iQ`"BM`v pHm2/bii2b y, R

*QMM2+iBQMKi`B+2b

k

"QHixKMM K+?BM2


ǳoBbB#H2Ǵ b2+iQ`

ǳ>B//2MǴ b2+iQ`

"BM`v pHm2/bii2b y, R

*QMM2+iBQMKi`B+2b

k

"QHixKMM K+?BM2


ǳoBbB#H2Ǵ b2+iQ`

ǳ>B//2MǴ b2+iQ`

"BM`v pHm2/bii2b y, R

*QMM2+iBQMKi`B+2b

Ç "QHixKMM K+?BM2 U"JV, h M/ Z = yXÇ _2bi`B+i2/ "QHixKMM K+?BM2 U_"JV, h = Z = yX

k

"QHixKMM K+?BM2

1M2`;v #b2/ KQ/2H, (>BMiQM- a2DMQrbFB ǵ3e)

oB2r b biiBbiB+H K2+?MB+H bvbi2KX

h?2 bvbi2K 2M2`;v 7Q` ;Bp2M bii2 p2+iQ`b (p, ?),

1(p, ?) = Rkpihp + R

k?iZ? + piq? + "?? + "pp

aii2 p2+iQ`b *QMM2+iBQM Ki`B+2b "Bb2b

j

"QHixKMM K+?BM2


oB2r b biiBbiB+H K2+?MB+H bvbi2KX

h?2 bvbi2K 2M2`;v 7Q` ;Bp2M bii2 p2+iQ`b (p, ?),


k?iZ? + piq? + "?? + "pp

aii2 p2+iQ`b *QMM2+iBQM Ki`B+2b "Bb2bj

"QHixKMM K+?BM2


ai`iBM; 7`QK i?2 bvbi2K 2M2`;v 7Q` ;Bp2M bii2 p2+iQ`b (p, ?),


k?iZ? + piq? + "?? + "pp

h?2 +MQMB+H T`iBiBQM 7mM+iBQM Bb /2M2/ b,

w =∑

?,p2−1(p,?)

S`Q##BHBiv i?2 bvbi2K Bb BM bT2+B+ bii2 ;Bp2M #v "QHixKMM /Bbi`B#miBQM,

S(p, ?) = 2−1(p,?)

wrBi? K`;BMHBxiBQM,

S(p) = 2−6(p)

w6`22 2M2`;v

9

"QHixKMM K+?BM2

G2`MBM;, (>BMiQM- a2DMQrbFB ǵ3e)

h?2Q`2iB+HHv- ;2M2`H +QKTmi2K2/BmKXoB /DmbiBM; q,h,Z,"?,"p #H2iQ H2`M i?2 mM/2`HvBM; T`Q##BHBiv/Bbi`B#miBQM Q7 ;Bp2M /ib2iX

>Qr2p2`, T`+iB+HHv MQi 72bB#H26Q` TTHB+iBQMb QMHv _"Jb ?p2 #22M +QMbB/2`2/X

8

_B2KMM@h?2i "QHixKMM K+?BM2

>Qr iQ +?M;2 i?2 biimb [mQ\ (E`2~- aX*X- >;?B;?i- E?H2M ǵRd)E22T i?2 BMM2` b2+iQ` +QmTHBM;b MQM@i`BpBH- #mi i?2 K+?BM2 bQHp#H2\

*QMiBMmQmbpHm2/ ∈ R


S(p) ≡ KmHiB@p`Bi2 ;mbbBM UiQQ i`BpBHV

e




ǳZmMiBx2/Ǵ∈ Z

aQK2i?BM; BMi2`2biBM; ?TT2MblM/2` KBH/ +QMbi`BMib QM +QMM2+iBQM Ki`B+2b UTQbBiBp2 /2MBi2M2bb-XXXV

S(p) ≡√

/2i h(kπ)Lp

2− Rk pihp−"i

pp−"iph−R"p

θ("i? + piq|Z)

θ("i? − "i

ph−Rq|Z − qih−Rq)

*HQb2/ 7Q`K MHviB+ bQHmiBQM biBHH pBH#H25

d




ǳZmMiBx2/Ǵ∈ Z

aQK2i?BM; BMi2`2biBM; ?TT2MblM/2` KBH/ +QMbi`BMib QM +QMM2+iBQM Ki`B+2b UTQbBiBp2 /2MBi2M2bb-XXXV

S(p) ≡√

/2i h(kπ)Lp

2− Rk pihp−"i

pp−"iph−R"p

θ("i? + piq|Z)

θ("i? − "i


*HQb2/ 7Q`K MHviB+ bQHmiBQM biBHH pBH#H25d


_h"J (E`2~- aX*X- >;?B;?i- E?H2M ǵRd)LQp2H p2`v ;2M2`B+ T`Q##BHBiv /2MbBiv,

S(p) ≡√

/2i h(kπ)Lp

2− Rk pihp−"i

pp−"iph−R"p

θ("i? + piq|Z)

θ("i? − "i


.KTBM; 7+iQ` _B2KMM@h?2i 7mM+iBQM

h?2 _B2KMM@h?2i /2MBiBQM,

θ(x,Ω) :=∑

M∈ZL?

2kπB( Rk MiΩM+Mix)

E2v T`QT2`iB2b, S2`BQ/B+Biv- KQ/mH` BMp`BM+2- bQHmiBQM iQ ?2i2[miBQM- 2i+XLQi2, :`/B2Mib +M #2 +H+mHi2/ MHviB+HHv b r2HH bQ ;`/B2Mi/2b+2Mi +M #2 mb2/ 7Q` QTiBKBxiBQMX

3

_h"J T`QT2`iB2b

q2 Q#b2`p2 i?i S(p) bivb BM i?2 bK2 /Bbi`B#miBQM mM/2` M2i`Mb7Q`KiBQMb- BX2X `QiiBQM M/ i`MbHiBQM

r = p + #, r ∼ S,#(p),

B7 i?2 HBM2` i`Mb7Q`KiBQM ?b 7mHH +QHmKM `MFXS,#(p) Bb i?2 /Bbi`B#miBQM S(p) rBi? T`K2i2`b `Qii2/ b

h−R → h−Ri , "p → (+)i"p − h# ,q → (+)iq , "? → "? − qi# .

r?2`2 + Bb i?2 H27i Tb2m/Q@BMp2`b2 /2M2/ b

+ = (i)−Ri.

N

TTHB+iBQMb

AKTH2K2MiiBQM

AM Q`/2` iQ T2`7Q`K i2bib r2 T`2T`2/ Tm#HB+ _h"J 7`K2rQ`F,h?2i, Svi?QM K+?BM2 H2`MBM; 7`K2rQ`F 7Q` _h"Jb M/ hLLb UrBi??2pv HB7iBM; /QM2 #v MmKTv- +vi?QM M/ *V

(?iiT,ff`B2KMMXBfi?2i)

Ç 1bv BMi2`7+2, E2`b HBF2/2MBiBQM Q7 KQ/2HX

Ç a:. M/ ;2M2iB+ QTiBKBx2` QmiQ7 i?2 #QtX1bv BMi2;`iBQM Q7 +mbiQKQTiBKBx2`bX

Ç 1bv iQ 2ti2M/ 7mM+iBQMHBivUQ#D2+i Q`B2Mi2/V

Ç *Sl #b2/ U7Q` +m``2Mi p2`bBQMV

Ry

http://riemann.ai/theta


AM i?2 M2ti r2 b?Qr 2tKTH2b Q7 _h"Jb 7Q`

Ç S`Q##BHBiv /2i2`KBMiBQMÇ .i +HbbB+iBQMÇ .i `2;`2bbBQMÇ aKTHBM;

RR


_h"J S(p) 2tKTH2b, (E`2~- aX*X- >;?B;?i- E?H2M ǵRd)

6Q` /Bz2`2Mi +?QB+2b Q7 T`K2i2`b UrBi? ?B//2M b2+iQ` BM R. Q` k.VX

Rk


JBtim`2 KQ/2H, (E`2~- aX*X- >;?B;?i- E?H2M ǵRd)

1tT2+iiBQM,b HQM; b i?2 /2MbBiv Bb r2HH 2MQm;?#2?p2/ i i?2 #QmM/`B2b Bi +M #2H2`M2/ #v M _h"J KBtim`2 KQ/2HX

Rj


1tKTH2b, (E`2~- aX*X- >;?B;?i- E?H2M ǵRd)

hQT Lp = R, L? = j, k, j- #miiQM Lp = k, L? = R (kt _h"J), kX R9


62im`2 /2i2+iQ`, (E`2~- aX*X- >;?B;?i- E?H2M ǵRd)aBKBH` iQ (E`Bx?2pbFv ǵyN)

L2r,*QM/BiBQMH 2tT2+iiBQMb Q7 ?B//2Mbii2b 7i2` i`BMBM;

1(?B|p) = − RkπB

∇Bθ(piq + "i?|Z)

θ(piq + "i?|Z)

h?2 /2i2+iQ` Bb i`BM2/ BM T`Q##BHBivKQ/2 M/ ;2M2`i2b 72im`2 p2+iQ`X

R8

62im`2 /2i2+iQ` 2tKTH2 @ D2i +HbbB+iBQM

C2i +HbbB+iBQM, (E`2~- aX*X- >;?B;?i- E?H2M ǵRd).i 7`QK ("H/B 2i HX ǵRe- ReyjXyNj9N)

.2b+`BKBMiBM; D2ib 7`QK bBM;H2?/`QMB+ T`iB+H2b M/ Qp2`HTTBM;D2ib 7`QK TB`b Q7 +QHHBKi2/?/`QMB+ T`iB+H2bX

.i UBK;2b Q7 jktjk TBt2HbV

Ç 8yyy BK;2b 7Q` i`BMBM;Ç k8yy BK;2b 7Q` i2biBM;

*HbbB2` h2bi /ib2i T`2+BbBQM

GQ;BbiB+ `2;`2bbBQM UG_V ddW_h"J 72im`2 /2i2+iQ` Y G_ 3jW

Re


h?2i L2m`H L2irQ`F, (E`2~- aX*X- >;?B;?i- E?H2M ǵRd)A/2,lb2 b +iBpiBQM 7mM+iBQM BM biM/`/ LLX h?2 T`iB+mH` 7Q`K Q7MQM@HBM2`Biv Bb H2`M2/ 7`QK /iX

E2v TQBMi,bKHH2` M2irQ`Fb M22/2/ #mi_B2KMM@h?2i 2pHmiBQM Bb 2tT2MbBp2X1tKTH2 UR,j@j@k,RV,

v(i) = y.yki + y.8 bBM(i + y.R) + y.d8 +Qb(y.k8i − y.j) +N (y, R)

v(i) hLL i hLL +iBpiBQMb Rd

_h"J bKTHBM; H;Q`Bi?Kh?2 T`Q##BHBiv 7Q` i?2 pBbB#H2 b2+iQ` +M #2 2tT`2bb2/ b,

S(p) =∑

[?]S(p|?)S(?)

r?2`2 S(p|?) Bb KmHiBp`Bi2 ;mbbBMX h?2 S(p)bKTHBM; +M #2 T2`7Q`K2/ 2bBHv #v,

Ç bKTHBM; ? ∼ S(?) mbBM; i?2 _h MmK2`B+H2pHmiBQM θ = θM + ϵ(_) rBi? 2HHBTbQB/ `/Bmb _ bQ

T =ϵ(_)

θM + ϵ(_)≪ R

Bb i?2 T`Q##BHBiv i?i TQBMi Bb bKTH2/ QmibB/2 i?22HHBTbQB/ Q7 `/Bmb _- r?BH2

∑

[?](_)

S(?) = θMθM + ϵ(_)

≈ R

BX2X bmK Qp2` i?2 HiiB+2 TQBMib BMbB/2 i?2 2HHBTbQB/XÇ i?2M bKTHBM; p ∼ S(p|?) R3

aKTHBM; 2tKTH2b

_h"J S(p) bKTHBM; 2tKTH2b, (aX*X M/ E`2~ ǵR3)

hQT Lp = R, L? = k, j (kt _h"J), j- #miiQM Lp = R, L? = jX

RN

aKTHBM; /BbiM+2 2biBKiQ`b

ky

aKTHBM; 2tKTH2b rBi? M2 i`Mb7Q`KiBQM

_h"J S(p) bKTHBM; rBi? M2 i`Mb7Q`KiBQM, (aX*X M/ E`2~ ǵR3)

6Q` `QiiBQM Q7 θ = π/9 M/ b+HBM; Q7 k ULp = k, L? = kVX

kR

*QM+HmbBQM

PmiHQQF

AM bmKK`v,

Ç L2r "J `+?Bi2+im`2 #b2/ QM i?2 _B2KMM@h?2i 7mM+iBQMXÇ _2bmHib `2 2M+Qm`;BM;- b2p2`H TTHB+iBQM QTTQ`imMBiB2bX

6Q` i?2 7mim`2,

Ç S2`7Q`K bvbi2KiB+ #2M+?K`FbXÇ .2p2HQT #2ii2` QTiBKBxiBQM H;Q`Bi?KbXÇ S`QpB/2 KQ`2 +QKTH2i2 T?vbB+b BMi2`T`2iiBQM UB7 TQbbB#H2V

kk

h?MF vQm5

kk

hbbm;?m qq`fb?qtqmjgbm:2qk2i`v m/s?vbb+b-rr ......t`2/b+ibqmb7q`b2p2` ht`q+2bb2b ig>*x ⇒...

Documents