Evidence for a doubly charm tetraquark pole with lattice QCD

Evidence for a doubly charm tetraquark pole with lattice QCD

Sasa Prelovsek

Faculty of Mathematics and Physics, University of Ljubljana Jozef Stefan Institute, Ljubljana , Slovenia

F1, IJS, 24

March 2022

based on 2202.101101

done in collaboration with M. Padmanath

0

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cc u ¯ d ¯

LHCb discovery of T _cc

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m(D

D

⇡

) [GeV]

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cc u ¯ d ¯

D⁰D^*+

The longest lived discovered hadron with explicitly exotic quark content

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Re(E) m_D^⇤₊ m_D⁰ [GeV]

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D

D

Pole in T(E)

m = 0.36(4) GeV

D*⁰D⁺

LHCb July 2021, 2109.01038, 2109.01056

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1 2

I=0, J

=1

Theoretical predictions

2

Theoretical PREdictions

courtesy: Ivan Polyakov, EPS-HEP 2021 (references at the back)

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m [MeV] references at the back

D*

D

v Phenomenological approaches

*

V_ijbetween all pairs of quarks, ground state energy of four-body problem

v Lattice QCD

only finite-volume eigen-energy En(L) was extracted:

this does not suffice to establish a near-threshold state

Junnarkar, Mathur, Padmanath, PRD 99, 034507 (2019), 1810.12285 Hadron Spectrum, JHEP 11, 033 (2017), 1709.01417

To establish a near-threshold state: pole in T(E) needs to be found:

T(E) has not been extracted by lattice QCD before our study

Our study

first and still the only extraction of T(E) with lattice QCD

pole related to Tccestablished for the first time with lattice QCD

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m = 1.6 ± 1.0 MeV

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T(E)/ 1

s m² = 1 E² m² E=E_cm

scattering amplitude T(E)

Summary of our lattice results ^m

^{> m}

m

≈ 280 MeV m

≈ 1927 MeV

lat

LHCb

Pole of T(E)

increasing m

Sketch of expected binding energy v T(E) extracted via the Luscher’s method v Evidence for pole related to Tcc

v For m

> m

one expects decreased attraction T

: bound state becomse virtual bound state indeed this is what we find

omitting

D⁰D⁰⇡⁺

Definitions: bound state, virtual bound state & resonance

4

Poles of T(E), ^E=E

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T (E) / 1 E

m

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T (E) / 1

E

m

+ iE

Bound st. Resonance p = i |p|

Virtual bound st.

p = - i |p|

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E = q

m

+ p

+ q

m

+ ( p)

< m

+ m

How did we arrive at these lattice QCD result ?

eigen-energies on the lattice -> T(E)

Lattice QCD ensembles employed

6

CLS Consortium

with dynamical quarks: u,d,s

m

=m

> m

, m

≈ 280 MeV Clover Wilson fermions

C = ∫ DG Dq Dq C e ^{− SQCD / !}

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S

= Z

d

x

L

(m

, g

)

L = 2.1 fm, 2.7 fm a ≈ 0.086 fm

Eucledian space-time

strategy:

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C ! E ! T (E )

Energies of DD* in non-interacting limit

periodic bc in space

P=0

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~

p

= ~ n

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E = q

m²_D +~p₁² + q

m²_D⇤ +~p₂²

D(0) D*(0) D(-1) D*(1)

E= q

m²_D+ (^2⇡_L)²+ q

m²_D⇤+ (^2⇡_L)²

L

Energies of DD* in non-interacting limit

8

L = 2.1 fm, 2.7 fm periodic bc in space

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E

⌘ m

+m

P=0

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~

p

= ~ n

<latexit sha1_base64="lkVkEWi/jFsd1s8q8kHlhUODraQ=">AAACJ3icbZDLSsNAFIYnXmu9RV26GSyCWChJKOpGKVrBZQV7gV7CZDpth84kcWZSKCE+jRtfxY2gIrr0TUzaitp6YODn/87hzPkdn1GpDONDm5tfWFxaTq2kV9fWNzb1re2K9AKBSRl7zBM1B0nCqEvKiipGar4giDuMVJ3+RcKrAyIk9dwbNfRJk6OuSzsUIxVbtn4GL08b8laokNvFlpVtDAiGvm22wjsrirLfKCy2DqMfbI0xTNt6xsgZo4KzwpyIDJhUydafG20PB5y4CjMkZd00fNUMkVAUMxKlG4EkPsJ91CX1WLqIE9kMR3dGcD922rDjifi5Co7c3xMh4lIOuRN3cqR6cpol5n+sHqjOSTOkrh8o4uLxok7AoPJgEhpsU0GwYsNYICxo/FeIe0ggrOJokxDM6ZNnRcXKmUe5/HU+UzifxJECu2APHAATHIMCuAIlUAYY3INH8AJetQftSXvT3setc9pkZgf8Ke3zCzHCpOo=</latexit>

E = q

m²_D +~p₁² + q

m²_D⇤ +~p₂²

D(0) D*(0) D(-1) D*(1)

E= q

m²_D+ (^2⇡_L)²+ q

m²_D⇤+ (^2⇡_L)²

L

Extracting eigen-energies from correlation functions

<latexit sha1_base64="CW6EyRP+fc57NpGN0xAIVa6bRbo=">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</latexit>

O = (¯ u

c)

( ¯ d

c)

(~ p

$ ~ p

)

<latexit sha1_base64="CFDdIQKkGtZawSfXNGg1DBzDFrc=">AAACO3icbVBNS8NAFNz4WetX1KOXxSLUS0lK/TiKXjxWsSo0Ibxst+3S3STsboQS4u/y4p/w5sWLB0W8endjK6jtwMIwM499b8KEM6Ud58mamZ2bX1gsLZWXV1bX1u2NzSsVp5LQFol5LG9CUJSziLY005zeJJKCCDm9DgenhX99S6VicXSphwn1BfQi1mUEtJEC+wJXvRAkTrHXAyEg2P8hGpO9IPNuKcFJ4OZ3o1znx2bTYvUc43JgV5ya8w08SdwxqaAxmoH96HVikgoaacJBqbbrJNrPQGpGOM3LXqpoAmQAPdo2NAJBlZ99357jXaN0cDeW5kVmmUL9PZGBUGooQpMUoPvqv1eI07x2qrtHfsaiJNU0IqOPuinHOsZFkbjDJCWaDw0BIpnZFZM+SCDa1F2U4P4/eZJc1WvuQa1x3qgcn4zrKKFttIOqyEWH6BidoSZqIYLu0TN6RW/Wg/VivVsfo+iMNZ7ZQn9gfX4BB8WrtA==</latexit>

(¯ u

c)

( ¯ d

c)

D D*

<latexit sha1_base64="SrCuB4Dln4QkPumAJpk7hUquBgI=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0VwUUpSiroRim5cuKhgH9CEMplO2qGTSZiZFEqIH+HGX3HjQhG3gjv/xmmahbYeGDj3nHu5c48XMSqVZX0bhZXVtfWN4mZpa3tnd8/cP2jLMBaYtHDIQtH1kCSMctJSVDHSjQRBgcdIxxtfz/zOhAhJQ36vphFxAzTk1KcYKS31zQqEzoRgGPUTu1JLL7OCzwv44ChfIJzUnIimyW3aN8tW1coAl4mdkzLI0eybX84gxHFAuMIMSdmzrUi5CRKKYkbSkhNLEiE8RkPS05SjgEg3ya5K4YlWBtAPhX5cwUz9PZGgQMpp4OnOAKmRXPRm4n9eL1b+hZtQHsWKcDxf5McMqhDOIoIDKghWbKoJwoLqv0I8QjoIpYMs6RDsxZOXSbtWtc+q9bt6uXGVx1EER+AYnAIbnIMGuAFN0AIYPIJn8ArejCfjxXg3PuatBSOfOQR/YHz+AEVznbQ=</latexit>

~

p

= ~ n

Diquark antidiquark operators [cc][ud] not incorporated: Emmanuel, Luka

J

=1

, I=0

t t=0

Energies of DD* from lattice QCD

10

L = 2.1 fm, 2.7 fm

P=0, J ^P =1 ⁺

D(0) D*(0) D(-1) D*(1)

L

v energies shifted from non-interacting energies:

renders info on T(E)

v focus on energy region near DD* threshold v scattering in partial wave l=2 negligible

[2]: DD* with J^P=1⁺ in l=0,2 :

both E degenerate in noninteracting limit

<latexit sha1_base64="FFHoxIDRiaNlzTNrihl4+tahnP0=">AAAB9HicbVBNSwMxEM36WetX1aOXYBHqpeyKqMeiFDxW6Bd0l5JNs21oNlmT2UIp/R1ePCji1R/jzX9j2u5BWx8MPN6bYWZemAhuwHW/nbX1jc2t7dxOfndv/+CwcHTcNCrVlDWoEkq3Q2KY4JI1gINg7UQzEoeCtcLh/cxvjZg2XMk6jBMWxKQvecQpASsFVR/7oHyM66XqRbdQdMvuHHiVeBkpogy1buHL7ymaxkwCFcSYjucmEEyIBk4Fm+b91LCE0CHps46lksTMBJP50VN8bpUejpS2JQHP1d8TExIbM45D2xkTGJhlbyb+53VSiG6DCZdJCkzSxaIoFRgUniWAe1wzCmJsCaGa21sxHRBNKNic8jYEb/nlVdK8LHvXZe/xqli5y+LIoVN0hkrIQzeogh5QDTUQRU/oGb2iN2fkvDjvzseidc3JZk7QHzifP1w0kIg=</latexit>

E ! T (E )

Luscher’s relation

Relation between E and δ(E), T(E): 1D quantum mechanics

periodic boundary condition

relation between δ, L and p or E x=R

E=p

/2m

V= 0: outside the region of potential

<latexit sha1_base64="4spDFslKCV1w5HO+fv+Z36bAyJM=">AAACDnicbVC7SgNBFJ2Nrxhfq5Y2gyFgY9gNojZC0MbCIoJ5QDaE2clsMmR2dpm5K4QlX2Djr9hYKGJrbeffOElW0MQDF86ccy9z7/FjwTU4zpeVW1peWV3Lrxc2Nre2d+zdvYaOEkVZnUYiUi2faCa4ZHXgIFgrVoyEvmBNf3g18Zv3TGkeyTsYxawTkr7kAacEjNS1S/GFB4EiNK14MR+nN2N5/CNMXl6PCSBdu+iUnSnwInEzUkQZal370+tFNAmZBCqI1m3XiaGTEgWcCjYueIlmMaFD0mdtQyUJme6k03PGuGSUHg4iZUoCnqq/J1ISaj0KfdMZEhjoeW8i/ue1EwjOOymXcQJM0tlHQSIwRHiSDe5xxSiIkSGEKm52xXRATBZgEiyYENz5kxdJo1J2T8vu7UmxepnFkUcH6BAdIRedoSq6RjVURxQ9oCf0gl6tR+vZerPeZ605K5vZR39gfXwDBBKcug==</latexit>

p =

n

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p = ^2⇡ _L n

<latexit sha1_base64="lFmsWZ+VdWr5pedjxFOiIAGT2CE=">AAACAHicbZDLSgMxFIYz9VbrbdSFCzfRItRNmRFRl0U3LivYC3SGkkkzbWgmCUlGLEM3voobF4q49THc+Tam7Sy09YfAx3/O4eT8kWRUG8/7dgpLyyura8X10sbm1vaOu7vX1CJVmDSwYEK1I6QJo5w0DDWMtKUiKIkYaUXDm0m99UCUpoLfm5EkYYL6nMYUI2OtrnsQSE2Do0AqIY2wgIWuyMfTrlv2qt5UcBH8HMogV73rfgU9gdOEcIMZ0rrje9KEGVKGYkbGpSDVRCI8RH3SschRQnSYTQ8YwxPr9GAslH3cwKn7eyJDidajJLKdCTIDPV+bmP/VOqmJr8KMcpkawvFsUZwyaAScpAF7VBFs2MgCworav0I8QAphYzMr2RD8+ZMXoXlW9S+q/t15uXadx1EEh+AYVIAPLkEN3II6aAAMxuAZvII358l5cd6dj1lrwcln9sEfOZ8/FX2WEw==</latexit>

/ cos(px)

<latexit sha1_base64="EWS2X+ty+MGu8fberApzSdP09NQ=">AAACC3icbZDLSsNAFIYn9VbrLerSTdoiVISSiKjLohuXFewFmlAm00k7dJIZZk7EUrp346u4caGIW1/AnW/jtM1Cqz8MfPznHM6cP5ScaXDdLyu3tLyyupZfL2xsbm3v2Lt7TS1SRWiDCC5UO8SacpbQBjDgtC0VxXHIaSscXk3rrTuqNBPJLYwkDWLcT1jECAZjde2SLzXzi75UQoIwQISuyHu/eGy4Rzngo65ddqvuTM5f8DIoo0z1rv3p9wRJY5oA4VjrjudKCMZYASOcTgp+qqnEZIj7tGMwwTHVwXh2y8Q5NE7PiYQyLwFn5v6cGONY61Ecms4Yw0Av1qbmf7VOCtFFMGaJTIEmZL4oSrkDwpkG4/SYogT4yAAmipm/OmSAFSZg4iuYELzFk/9C86TqnVW9m9Ny7TKLI48OUAlVkIfOUQ1dozpqIIIe0BN6Qa/Wo/VsvVnv89aclc3so1+yPr4BfFuaDA==</latexit>

/ cos(px + )

12

Relation between E and 𝛿 𝐸 , 𝑇(𝐸)

H

(p) H

(-p)

E

= 𝑚

+ 𝑝

+ 𝑚

+ 𝑝

E = eigen-energy lattice from lattice in cmf Luscher’s relation :

Luscher 1991

<latexit sha1_base64="ttXtpubnF82ZjV5Mpw6WKc/yoKA=">AAACEXicbZDLSsNAFIYnXmu9RV26GSxCN5akFHUjFEVwWaE3aEKZTCft0MmFmROhhLyCG1/FjQtF3Lpz59s4bbPQ1h8GPv5zDmfO78WCK7Csb2NldW19Y7OwVdze2d3bNw8O2ypKJGUtGolIdj2imOAhawEHwbqxZCTwBOt445tpvfPApOJR2IRJzNyADEPuc0pAW32zjHHzyvEloeltllazOdpZGjs0AmfABBB8xuOsb5asijUTXgY7hxLK1eibX84goknAQqCCKNWzrRjclEjgVLCs6CSKxYSOyZD1NIYkYMpNZxdl+FQ7A+xHUr8Q8Mz9PZGSQKlJ4OnOgMBILdam5n+1XgL+pZvyME6AhXS+yE8EhghP48EDLhkFMdFAqOT6r5iOiM4EdIhFHYK9ePIytKsV+7xSu6+V6td5HAV0jE5QGdnoAtXRHWqgFqLoET2jV/RmPBkvxrvxMW9dMfKZI/RHxucPApudKw==</latexit>

T = E 2

1

p cot ip

l=0:

<latexit sha1_base64="bb3RpvRBF5CE1Aop9HQZ3e6a9wg=">AAACC3icbVDLSgNBEJyN7/iKevQyJAiCEHZDUC+CKIJHRfOA7BpmJ706ZPbBTK8Qlr178Ve8eFDEqz/gzb9xEnPQxIKGoqqb7i4/kUKjbX9ZhZnZufmFxaXi8srq2nppY7Op41RxaPBYxqrtMw1SRNBAgRLaiQIW+hJafv906LfuQWkRR9c4SMAL2W0kAsEZGqlbKtOrI2dPuBgoxrN6kmdn+fUR3GQ14fZAIsuL3VLFrtoj0GnijEmFjHHRLX26vZinIUTIJdO649gJehlTKLiEvOimGhLG++wWOoZGLATtZaNfcrpjlB4NYmUqQjpSf09kLNR6EPqmM2R4pye9ofif10kxOPQyESUpQsR/FgWppBjTYTC0JxRwlANDGFfC3Er5HTOpoIlvGIIz+fI0adaqzn61flmvHJ+M41gk26RMdolDDsgxOScXpEE4eSBP5IW8Wo/Ws/Vmvf+0FqzxzBb5A+vjG33Omhk=</latexit>

S = 1 + i ^4p _E T = e ²ⁱ

<latexit sha1_base64="KSq9lzFQZ7tHd/2WXI0UgzoE4jw=">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</latexit>

p cot (p) = 2 Z

(1, (

)

) p ⇡L

<latexit sha1_base64="j2C3xtZvXG+c5C8kKX4rjk/A0lw=">AAACAXicbZDLSsNAFIYn9VbrLepGcDNYhApSEhF1WZSCywq9QRPKZDJth04uzJwIJbQbX8WNC0Xc+hbufBunbRba+sPAx3/O4cz5vVhwBZb1beRWVtfWN/Kbha3tnd09c/+gqaJEUtagkYhk2yOKCR6yBnAQrB1LRgJPsJY3vJvWW49MKh6FdRjFzA1IP+Q9Tgloq2seVR3sQORgXC9Vz84njs8EEI1ds2iVrZnwMtgZFFGmWtf8cvyIJgELgQqiVMe2YnBTIoFTwcYFJ1EsJnRI+qyjMSQBU246u2CMT7Xj414k9QsBz9zfEykJlBoFnu4MCAzUYm1q/lfrJNC7cVMexgmwkM4X9RKBIcLTOLDPJaMgRhoIlVz/FdMBkYSCDq2gQ7AXT16G5kXZvirbD5fFym0WRx4doxNUQja6RhV0j2qogSiaoGf0it6MJ+PFeDc+5q05I5s5RH9kfP4ALtiUzQ==</latexit>

E ! T (E ), (E )

known function

DD* scattering amplitude with l =0 P=0, J ^P =1 ⁺

green and blue points

fit using just P=0

<latexit sha1_base64="FFHoxIDRiaNlzTNrihl4+tahnP0=">AAAB9HicbVBNSwMxEM36WetX1aOXYBHqpeyKqMeiFDxW6Bd0l5JNs21oNlmT2UIp/R1ePCji1R/jzX9j2u5BWx8MPN6bYWZemAhuwHW/nbX1jc2t7dxOfndv/+CwcHTcNCrVlDWoEkq3Q2KY4JI1gINg7UQzEoeCtcLh/cxvjZg2XMk6jBMWxKQvecQpASsFVR/7oHyM66XqRbdQdMvuHHiVeBkpogy1buHL7ymaxkwCFcSYjucmEEyIBk4Fm+b91LCE0CHps46lksTMBJP50VN8bpUejpS2JQHP1d8TExIbM45D2xkTGJhlbyb+53VSiG6DCZdJCkzSxaIoFRgUniWAe1wzCmJsCaGa21sxHRBNKNic8jYEb/nlVdK8LHvXZe/xqli5y+LIoVN0hkrIQzeogh5QDTUQRU/oGb2iN2fkvDjvzseidc3JZk7QHzifP1w0kIg=</latexit>

E ! T (E)

Luscher’s relation

DD* scattering amplitude with l =0

14

P=0, J ^P =1 ⁺

<latexit sha1_base64="1ITQFlDVmEZhUagqn75nZlfbTnc=">AAACDHicbVDLSgMxFM34rPVVdekmWIS6KTNSVIRC0Y24qtAXdMYhk2ba0MyD5I5QhvkAN/6KGxeKuPUD3Pk3po+Fth4IHM45l5t7vFhwBab5bSwtr6yurec28ptb2zu7hb39looSSVmTRiKSHY8oJnjImsBBsE4sGQk8wdre8Hrstx+YVDwKGzCKmROQfsh9TgloyS0UG24qqmZ2n5Zuq9ZJdoltHNs0ArvHBBA3NbOqTpllcwK8SKwZKaIZ6m7hy+5FNAlYCFQQpbqWGYOTEgmcCpbl7USxmNAh6bOupiEJmHLSyTEZPtZKD/uR1C8EPFF/T6QkUGoUeDoZEBioeW8s/ud1E/AvnJSHcQIspNNFfiIwRHjcDO5xySiIkSaESq7/iumASEJB95fXJVjzJy+S1mnZOitX7irF2tWsjhw6REeohCx0jmroBtVRE1H0iJ7RK3oznowX4934mEaXjNnMAfoD4/MHfhWaBw==</latexit>

T _l=0 ^{(J =1)} : p cot ₀ =

expansion near threshold

red line

E

(fit params)

fit using just P=0

DD* scattering amplitude with l =0

<latexit sha1_base64="ttXtpubnF82ZjV5Mpw6WKc/yoKA=">AAACEXicbZDLSsNAFIYnXmu9RV26GSxCN5akFHUjFEVwWaE3aEKZTCft0MmFmROhhLyCG1/FjQtF3Lpz59s4bbPQ1h8GPv5zDmfO78WCK7Csb2NldW19Y7OwVdze2d3bNw8O2ypKJGUtGolIdj2imOAhawEHwbqxZCTwBOt445tpvfPApOJR2IRJzNyADEPuc0pAW32zjHHzyvEloeltllazOdpZGjs0AmfABBB8xuOsb5asijUTXgY7hxLK1eibX84goknAQqCCKNWzrRjclEjgVLCs6CSKxYSOyZD1NIYkYMpNZxdl+FQ7A+xHUr8Q8Mz9PZGSQKlJ4OnOgMBILdam5n+1XgL+pZvyME6AhXS+yE8EhghP48EDLhkFMdFAqOT6r5iOiM4EdIhFHYK9ePIytKsV+7xSu6+V6td5HAV0jE5QGdnoAtXRHWqgFqLoET2jV/RmPBkvxrvxMW9dMfKZI/RHxucPApudKw==</latexit>

T = E 2

1

p cot ip

p cot 𝛿

virtual bound st. condition

i p =i(-i|p|)=|p|

virtual bound st.

pole of T(E)

fit using just P=0

16

DD* scattering amplitude with l =0,1

P=0 P=0 P=1

Predicting eigen-energies ^E

(fit params)

P=0 P=0 P=(0,0,1) P=(1,1,0) P=(0,0,2)

Location of T _cc pole

18

m

≈ 280 MeV m

≈ 1927 MeV m

≈ 2049 MeV

lat

LHCb

Pole of T(E)

v lat: evidence for virtual bound state pole v we expect this pole is related to

Tcc bound state pole found by LHCb : arguments in Supplement of 2202.101101 and in the following slides

Sketch of expected binding energy

increasing m

Expected dependence of T _cc on m _u/d : simple QM arguments

Yukava-like potential

exchanged particles:

light mesons 𝜋, 𝜌, . .

increasing m

increasing m

decreasing attraction |V|

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V (r) / e

r

Simplest Example: scattering in square-well potential in QM

20

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u(r) = Asin(qr)

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u(r) = Bsin(pr+ )

R r

increasing m

, decreasing attraction

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e

= e

p=i|p| p=-i|p|

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e^ipr=e^|p|r

DD* rate ∝ p|T|

ra te ∝ p| T|

for T extracted on lattice

All fully attractive potentials lead to analogous conclusions

video: courtesy M. Padmanath

Dependence on the charm quark mass

22

closer to physical (presented till now)

simulation at two charm-quak masses

DD* scattering amplitude with l =0,1

P=0 P=0 P=1

at m

≈1927 MeV

24

DD* scattering amplitude with l =0,1 ^{at m}

≈1762 MeV (lighter charm quark mass)

P=0 P=0 P=1

Lattice results at two m _c

Observed m

dependence in agreement with QM arguments for fully attractive potential

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V (r) = V

f (r/R) V(r) independent on m

,

reduced mass m

of D,D* system increases with m

26

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cc d ¯ u ¯

+

likely dominant

T _cc ^{(I=0, J}

⁼¹

⁾

Conclusions on doubly charm tetraquark

v The longest lived exotic hadron ever found v It lies very close to DD* threshold

v Lattice QCD:

to establish a state near threshold, scattering amplitude has to be extracted and pole identified

Our study 2202.101101 :

- the only extraction of DD* scattering amplitude - virtual bound state pole found at m

≈ 280 MeV - likely related to Tcc found by LHCb

Many interesting questions and quantities still to be explored ...

Backup

28

Previous lattice QCD study of T _cc channel

Junnarkar, Mathur, Padmanath, PRD 99, 034507 (2019), 1810.12285

lowest finite-volume eigen-energy for

P=0, J

=1

, I=0

30

Previous lattice QCD study of T _cc channel

finite-volume eigen-energies for

P=0, J

=1

, I=0

Hadron Spectrum, JHEP 11, 033 (2017), 1709.01417

Th"ory pr"dictions

Tcc theory predictions (taken from Ivan Polyakov, EPS meeting)

32

Tcc theory predictions (taken from Ivan Polyakov, EPS meeting)

Interpolators

Example: P=0

J

=1

-> cubic irrep T

P=D, V=D*

34

Relation between E and 𝛿 𝐸 , 𝑇(𝐸)

E

= 𝑚

+ 𝑝

+ 𝑚

+ 𝑝

E = eigen-energy lattice from lattice in cmf

Luscher’s relation (only l =0):

Luscher 1991 + generalizations

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T = E 2

1

p cot ip

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p cot ₀ = B (E, L)

known kinematical

function

lattice eigen-energy

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S = 1 + i ^4p _E T = e ²ⁱ

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p cot ₀ B (E, L) = 0

Luscher’s relation ( l =0,1):

even and odd l contribute to given irrep for nonzero mom.

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det

✓ p cot

0 0 p

cot

◆

B (E, L) = 0

known 2x2 matrix of kinematical

functions (non-diagonal)

s-wave scattering on spherical potential well

q = 2 µ ^{( V}

+ E ) = 2 µ ^V

+ p

A sin qr B sin( pr + δ

)

δ

^{( p)} = arctan p

q tan(qR)

⎛

⎝ ⎜ ⎞

⎠ ⎟ − pR + n π 𝑢 𝑅 = 𝐴 sin 𝑞𝑅 = 𝐵 sin(𝑝𝑅 + 𝛿)

𝑢′ 𝑅 = 𝑞 𝐴 cos 𝑞𝑅 = 𝑝 𝐵 cos(𝑝𝑅 + 𝛿)

1

𝑞 tan 𝑞𝑅 = 1

𝑝 tan(𝑝𝑅 + 𝛿)

dividing both eqs