Hidden Analytic Relations for Two-Loop Higgs Amplitudes in QCD

靳庆军（中科院理论物理所）

EFT & Amplitude Workshop @USTCSeptember 6-9, 2019

Based on work in collaboration with Gang Yang 1804.04653, 1904.07260

Content

• Motivations

• Computations

• The hidden relation

• Summary and outlook

2

Scattering Amplitudes in Standard Model (SM)

Higgs production in LHC and CEPC

Model Scattering Amplitude Experiment Data

LHC : Perfect agreement with SM.

Future CEPC : higher accuracy, less background noise. Requires even high loop computation of scattering amplitudes.

3

New Methods in Scattering Amplitudes

• New methods developed during the computation of scattering amplitudes in supersymmetric field thoeries: spinor helicityformalism, on shell unitarity cut …

• N=4 SYM integrand was computed

to 5 loops. [Bern et al 2012]

• N=8 supergravity is free of UV

divergence to at least 5 loops. [Bern et al 2018]

• Apply these new methods to the computation of loop amplitude in Standard Model.

4

Higgs to 3 parton amplitude and the Maximal Transcendental Principle

5

The maximally transcendental principle: the maximal transcendental part of N=4 SYM and QCD amplitudes are the same.

Study the properties of Higgs boson is one of the major tasksof LHC and future CEPC.

In LHC, Higgs can be produced by the fusion of gluons and quarks. Higgs to 3 parton amplitudes are very important to understand the LHC data.

An “easy” N=4 computation gives information for the “difficult” QCD amplitude!

Content

• Motivations

• Computations

• The hidden relation

• Summary and outlook

6

Higgs effective field theory (HEFT)

• When the transverse momentum of Higgs particle is less than the top quark mass, the HEFT [Wilczek 1977] is good approximation.

• The leading (dimension 4) contribution of Higgs to 3 parton amplitude was computed in [Gehrmann, Jaquier, Glover, Koukoutsakis 2011].

• The contributions of dimension 6 operators. [QJ, Yang 2018] and [QJ, Yang 2019].

Le↵ = C0O0 +1

m2t

4X

i=1

CiOi +O

✓1

m4t

◆

O0= HTr(F 2)

O1= HTr(F ⌫µ F ⇢

⌫ F µ⇢ ) , O2 = HTr(D⇢Fµ⌫D

⇢Fµ⌫) ,

O3= HTr(D⇢F⇢µD�F�µ) , O4 = HTr(Fµ⇢D

⇢D�F�µ) .

⇒ ⇒

HTr(F 2) HTr(F 3) · · ·

Integrate out the top quark

7

Feynman Diagram vs Unitarity Cut

Imposing cut condition

loop amplitude two on shell tree amplitudes

8

A

A

A

A

F

F

F

F

T1 G1 N1

A

A

A

A

y

y

y

y

T1 C1N2

A

A

A

A

y

y

y

y

T1 C2N3

A

A

A

A

U

U

U

U

T1 G2 N4

A

A

A

A

c

c

c

c

T1 C1N5

A

A

A

A

c

c

c

c

T1 C2N6

A

A

A

A

V

V

V

V

T1 G3 N7

A

A

A

A

A

A

A

A

T1 C1N8

A

A

A

A

F

F

F

F

T2 G1 N9

A

A

A

A

y

y

y

y

T2 C1N10

A

A

A

A

y

y

y

y

T2 C2N11

A

A

A

A

U

U

U

U

T2 G2 N12

A

A

A

A

c

c

c

c

T2 C1N13

A

A

A

A

c

c

c

c

T2 C2N14

A

A

A

A

V

V

V

V

T2 G3 N15

A

A

A

A

A

A

A

A

T2 C1N16

A

A

A

A

F FF

FT3 G1 N17

A

A

A

A

y yy

yT3 C1N18

A

A

A

A

y yy

yT3 C2N19

A

A

A

A

U UU

UT3 G2 N20

A

A

A

A

c cc

cT3 C1N21

A

A

A

A

c cc

cT3 C2N22

A

A

A

A

V VV

VT3 G3 N23

A

A

A

A

A AA

AT3 C1N24

A A Æ A A

Feynman diagrams: a broken vase

D-dimensional unitarity cut

• 4-d spinor helicity formalism fails to capture the rational term in non-SUSY theories.

• D-dimensional unitarity uses the D-dimensional tree amplitude.

• The polarization vector summing rule:

9

A���� = (2� 2✏)µ4 =4

3+O(✏)

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A(✏1, ✏2, ✏3, ✏4) =� (✏1 · ✏4)(✏2 · ✏3) + (✏1 · ✏3)(✏2 · ✏4)� (✏1 · ✏2)(✏3 · ✏4))

� 2

s

h2t(✏1 · ✏2)(✏3 · ✏4) + (✏1 · p3)(✏2 · p1)(✏3 · ✏4)� (✏1 · p2)(✏2 · p3)(✏3 · ✏4)

i

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X

helicities

"µi "⌫i = ⌘µ⌫ � qµp⌫i + q⌫pµi

q · pi<latexit sha1_base64="Fg2m3TvnOOH7Xd9mRE15ylT1t7k=">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</latexit>

New Strategy of IBP

10

Loop integrand can be reduced using integration by parts (IBP) relations.

The IBP with cut strategy enhanced the efficiency by a factor of 10.

Simplify the multi-polylog functions using Symbol

11

Content

• Motivations

• Computations

• The hidden relation

• Summary and outlook

12

Transcendental Degree

• deg=0: Algebraic number, ; rational functions,

• deg=1:

• deg=k:

13

2�p3

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uv

w2<latexit sha1_base64="L9hcpC5cv6jTQr0wDAWlnwDHfAM=">AAAB9XicbVBNS8NAEJ3Ur1q/oh69LBbBU0mqoMeiF48V7Ae0adlsN+3SzSbsblpKyP/w4kERr/4Xb/4bt20O2vpg4PHeDDPz/JgzpR3n2ypsbG5t7xR3S3v7B4dH9vFJU0WJJLRBIh7Jto8V5UzQhmaa03YsKQ59Tlv++H7utyZUKhaJJz2LqRfioWABI1gbqdcNJCZpMsnSaa+a9e2yU3EWQOvEzUkZctT79ld3EJEkpEITjpXquE6svRRLzQinWambKBpjMsZD2jFU4JAqL11cnaELowxQEElTQqOF+nsixaFSs9A3nSHWI7XqzcX/vE6ig1svZSJONBVkuShIONIRmkeABkxSovnMEEwkM7ciMsImCG2CKpkQ3NWX10mzWnGvKtXH63LtLo+jCGdwDpfgwg3U4AHq0AACEp7hFd6sqfVivVsfy9aClc+cwh9Ynz8XApLk</latexit>

⇣k, Lik(u

v)

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⇡, log(1� u� v),1

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D = 4� 2✏<latexit sha1_base64="3NhKawYQ934VtPB3W8MemLD9nys=">AAACB3icbVA9SwNBEN3zM8avqKUgh0GwMdzFgDZCUAvLCOYDciHsbSbJkr29Y3dODEc6G/+KjYUitv4FO/+Nm0sKTXww8HhvZmfn+ZHgGh3n21pYXFpeWc2sZdc3Nre2czu7NR3GikGVhSJUDZ9qEFxCFTkKaEQKaOALqPuDq7FfvweleSjvcBhBK6A9ybucUTRSO3fgITxg+k6ioDNKri9KJ0UPIs1FKEftXN4pOCnseeJOSZ5MUWnnvrxOyOIAJDJBtW66ToSthCrkTMAo68UaIsoGtAdNQyUNQLeSdP/IPjJKx+6GypREO1V/TyQ00HoY+KYzoNjXs95Y/M9rxtg9byVcRjGCZJNF3VjYGNrjUOwOV8BQDA2hTHHzV5v1qaIMTXRZE4I7e/I8qRUL7mmheFvKly+ncWTIPjkkx8QlZ6RMbkiFVAkjj+SZvJI368l6sd6tj0nrgjWd2SN/YH3+AHsambA=</latexit>

This definition allows us to discuss the transcendental degree of UV

or IR divergent quantities

deg⇣⇡2⇣3 log u

✏2

⌘= 8

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The maximally transcendental principle

• The MT part of N=4 and QCD amplitude are the same?

• The MT part of amplitude is universal in all gauge theories?

• Anomalous dimensions of twist-two operators [Kotikov, Lipatov, Onishchenko, Velizhanin 2004]

• The dimension 4 contribution of Higgs to 3 gluon ( ) amplitude in HEFT [Brandhuber, Travaglini, Yang 2012].

H ! 3g

14

The MTP of the IR divergence

• IR divergence can be determined by cusp and jet anomalous dimensions.

• The MT part of cusp anomalous dimension is theory-independent up to 3 loops.

15

lnZ ⇠ ↵(�cusp1

4✏2+

�q,g1

2✏) + ↵2(

�cusp2

16✏2+

�q,g2

4✏) + ↵3(

�cusp3

36✏2+

�q,g3

6✏)

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IR renormalization factor

The MTP of the IR divergence

• The MT part of jet anomalous dimension only depends on the group representation of the particle up to 3 loops.

• MTP of IR divergence:

16

�q���MT

CF ! CA������! �g���MT

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�g1 =� 11

3CA +

4

3TFnf ,

�g2 =C2

A

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27+

11⇡2

18+2⇣3

◆+ CATFnf

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27� 2⇡2

9

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�g3 =C3

A

✓�97186

729+

6109⇡2

486� 319⇡4

270+

122

3⇣3�

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9⇣3 � 16⇣5

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243+

82⇡4

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712

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9⇣3

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FTFnf

+ CAT2Fn

2f

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729+

40⇡2

81� 224

27⇣3

◆� 44

9CFT

2Fn

2f .

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�q1 =� 3CF ,

�q2 =C2

F

✓�3

2+ 2⇡2�24⇣3

◆+ CFCA

✓�961

54� 11⇡2

6+26⇣3

◆+ CFTFnf

✓130

27+

2⇡2

3

◆,

�q3 =C3

F

✓�29

2� 3⇡2 � 8⇡4

5� 68⇣3+

16⇡2

3⇣3 + 240⇣5

◆

+ C2FCA

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4+

205⇡2

9+

247⇡4

135� 844

3⇣3�

8⇡2

3⇣3 � 120⇣5

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+ CFC2A

✓�139345

2916� 7163⇡2

486� 83⇡4

90+

3526

9⇣3�

44⇡2

9⇣3 � 136⇣5

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+ C2FTFnf

✓2953

27� 26⇡2

9� 28⇡4

27+

512

9⇣3

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+ CFCATFnf

✓�17318

729+

2594⇡2

243+

22⇡4

45� 1928

27⇣3

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+ CFT2Fn

2f

✓9668

729� 40⇡2

27� 32

27⇣3

◆.

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Hidden relation in amplitude

• The dimension 4 contribution of and amplitudes are given in [Gehrmann et al 2011].

• We rewrite the expressions using quadratic Casmir:

H ! 3g

H ! qqg

H ! qqg

CA = N, CF =N2 � 1

2N<latexit sha1_base64="2JVC/LghLYh5kOybk5KfWoarIZM=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBhZYkCropVAviqlSwD2hqmEwn7dDJg5mJUEI2bvwVNy4Uces/uPNvnLZZaOuBC4dz7uXee9yIUSEN41vLLSwuLa/kVwtr6xubW/r2TlOEMcekgUMW8raLBGE0IA1JJSPtiBPku4y03GF17LceCBc0DO7kKCJdH/UD6lGMpJIcfb/qXJZrxzasOtdl2+MIJ7V768RME6uWOnrRKBkTwHliZqQIMtQd/cvuhTj2SSAxQ0J0TCOS3QRxSTEjacGOBYkQHqI+6SgaIJ+IbjL5IoWHSulBL+SqAgkn6u+JBPlCjHxXdfpIDsSsNxb/8zqx9C66CQ2iWJIATxd5MYMyhONIYI9ygiUbKYIwp+pWiAdIRSFVcAUVgjn78jxpWiXztGTdnhUrV1kcebAHDsARMME5qIAbUAcNgMEjeAav4E170l60d+1j2prTspld8Afa5w92AZam</latexit>

H ! qqg

amplitude reduces to amplitude by replacing quadratic Casmir from Fundamental to Adjoint. [QJ, Yang 2019]

H ! 3gCF ! CA

H ! qqg H ! 3g

17

Verify MTP for dimension-6 operators

18

H ! 3g : Verified [QJ, Yang 2018]

H ! qqg

The same relation as the dimension-4 operator was verified. [QJ, Yang 2019]

H ! 3gCF ! CA

The Universality of MTP

• Changed the effective interaction between Higgs and partons, including

• Replaced quark by scalar fields and computed amplitudes.

• MTP is universal. It does not depend on the spin and is insensitive to the details of interaction.

19

H , HF3, HFD

2F

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H ! ��g<latexit sha1_base64="h/C3FX09K5rYdzPuHo1HkwGvXwU=">AAACBXicbZC7TsMwFIadcivlFmCEwaJCYqqSggRjBUvHItGL1ESV4zqpVceObAdURV1YeBUWBhBi5R3YeBucNgO0/JKlT/85R8fnDxJGlXacb6u0srq2vlHerGxt7+zu2fsHHSVSiUkbCyZkL0CKMMpJW1PNSC+RBMUBI91gfJPXu/dEKir4nZ4kxI9RxGlIMdLGGtjHTU/SaKSRlOIBesmIegGSWQ7TaGBXnZozE1wGt4AqKNQa2F/eUOA0JlxjhpTqu06i/QxJTTEj04qXKpIgPEYR6RvkKCbKz2ZXTOGpcYYwFNI8ruHM/T2RoVipSRyYzhjpkVqs5eZ/tX6qwys/ozxJNeF4vihMGdQC5pHAIZUEazYxgLCk5q8Qj5BEWJvgKiYEd/HkZejUa+55rX57UW1cF3GUwRE4AWfABZegAZqgBdoAg0fwDF7Bm/VkvVjv1se8tWQVM4fgj6zPHyb7mP4=</latexit>

Content

• Motivations

• Computations

• The hidden relation

• Summary and outlook

20

Summary and outlook

• Analytical Higgs to 3 parton amplitude at two loops.

• Extend MTP to amplitude.

• The origin of MTP?

• Any relation between MT part of N=4 SYM and QCD gluon amplitudes?

H ! qqg

21

Thank you!

22