Joannis Papavassiliou Department of Theoretical Physics and IFIC University of Valencia-CSIC

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1 The dynamics of the gluon mass Joannis Papavassiliou Department of Theoretical Physics and IFIC University of Valencia-CSIC Worshop on Functional Methods in Hadron and Nuclear Physics 2-25 August, 27 ECT*

2 Large volume lattice simulations The gluon propagator saturates in the deep infrared, both in the Landau gauge and away from it I.L.Bogolubsy, et al, PoS LAT27, 29 (27) P. Bicudo et al, Phys. Rev.D 92, 454 (25) Δ( )[ - ] β= = β= = β= = β= = Δ( )[ - ] ξ= ξ= ξ= ξ= ξ= ξ= [ ] [ ]

3 D. Binosi, L. Chang, J. Papavassiliou, and C. D. Roberts, Phys. Lett.B742, 83 (25). A. Cucchieri and T. Mendes, PoS LAT27, 297 (27), I. Bogolubsy, E. Ilgenfritz, M. Muller-Preusser, and A. Sternbec, Phys. Lett. B676, 69 (29). O. Oliveira and P. Silva, PoS LAT29, 226 (29). P. O. Bowman et al., Phys. Rev. D76, 9455 (27). A. C. Aguilar and A. A. Natale, JHEP 8, 57 (24). A. C. Aguilar, D. Binosi, and J. Papavassiliou, Phys. Rev. D78, 25 (28). P. Boucaud et al., JHEP 6, 99 (28), C. S. Fischer, A. Maas, and J. M. Pawlowsi, Annals Phys. 324, 248 (29). D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersicel, and H. Verschelde, Phys. Rev. D78, 6547 (28). J. Braun, H. Gies, and J. M. Pawlowsi, Phys. Lett. B684, 262 (2). M. Pennington and D. Wilson, Phys. Rev. D84, 99 (2). K.-I. Kondo, Phys. Rev.D74, 253 (26) ; Rev. D84, 672 (2). J. Rodriguez-Quintero, JHEP, 5 (2). J. Serreau and M. Tissier, Phys. Lett. B72, 97 (22).

4 aturation can be explained through dynamical gluon mass generation J. M. Cornwall, Phys. Rev. D26, 453 (982) running mass m 2 (q 2 ) Saturation: () = m 2 () Protects (most) perturbative logs: ln(q 2 )! ln(q 2 + m 2 (q 2 )) hard mass Strong Dynamics and Emergent Mass Daniele Binosi, Joannis Papavassiliou, and Craig D. Roberts Joint contribution for 2nd worshop on The Proton Mass (ECT*) Emergence: low-level rules producing high-level phenomena, with enormous apparent complexity TNT V at ECT*

5 Gluon SDE in the PT-BFM framewor 3. (q 2 )P µ (q) = q2 P µ (q)+i e µ (q) +G(q 2 ) P µ (q) =g µ q µ q q 2 Known function In Landau gauge q µ q q µ e µ (q, r, p) =i (r) i (p) q µ e µ (q, r, p) =D (p) D (r) q µ emnrs µ = f mse f ern + f mne f esr + f mre f ens Stronger version of the transversality of the gluon self-energy

6 Gluon propagator at the origin in the absence of poles q µ e µ (q, r, p) =i (r) i Taylor expansion of both sides around q = assuming no poles e µ (,r, r) = µ () () e µ (,, ) µ (p) /q 2 in the form factors of e µ (q, r, p) Plug into gluon self-energy () = lim q! Tr () = No gluon mass Seagull identity

7 Schwinger mechanism in Yang-Mills theories J.S. Schwinger, Phys. Rev.25, 397 (962); Phys.Rev.28, 2425 (962). To evade the previous result, one must relax one of the underlying assumptions. In particular, the derivation of the WI hinges on the absence of poles /q 2 Therefore, let us generate dynamically massless poles in e µ e µ (q, r, p) = enp µ (q, r, p)+ep µ (q, r, p) β,c p µ, a q r α, b = Γ np + i q 2 }{{} Γ p R. Jaciw and K. Johnson, Phys. Rev. D8, 2386 (973) E. Eichten and F. Feinberg, Phys. Rev. D, 3254 (974) Colored composite bound state excitation

8 The poles are longitudinally coupled e p µ (q, r, p) =q µ q 2 e C (q, r, p) contains 5 form factors Bose symmetry e C (,r, r) = Landau gauge: e C (q, r, p) = e C (q, r, p)g +... e C (,r, r) = ep (,r, q µ r) =lim C e q! q 2 (q, r, r =2r e ) C (q, r, r q) (r + q) 2 ec (r 2 ) q= q=

9 Ward identities in the presence of poles e µ (q, r, p) = enp µ (q, r, p)+q µ q 2 e C (q, r, p) Same ST identity! q µ e µ (q, r, p) =i (r) i (p) q µ e np µ (q, r, p)+ e C (q, r, p) =i (r) i (p), Expand around q = µ (,r, r) = i@ µ µ e C (q, r, p) q=

10 Evading the seagull identity () = lim q! Tr Triggers seagull identity exactly as before Vanishes identically () Z 2 2 ( 2 ) e C ( 2 ) A.C.Aguilar, D.Binosi, C.T.Figueiredo and J.P., Phys. Rev. D 94, no. 4, 452 (26)

11 Dynamical formation of massless poles Bethe-Salpeter equation for the full vertex q p r = + K K Substitute: e = enp + R ep q! Equate terms inear in q ρ, c p q = q = = a a p ν, b β α + q σ γ K K ρ, c p p ν, b Bethe-Salpeter equation for massless pole formation

12 Linear and homogeneous integral equation Solution of the BSE ec (p 2 )= Z ec ( 2 ) 2 () ( + p) K(, p) Eigenvalue proportional to s ρ,b α,m ρ,b α,m ρ,b α,m K K + q r r + q + + q r r + + r + q r r + q γ σ,c ( ) β,n σ,c r + q ( ) β,n σ,c + q ( ) β,n 2. Solution determined up to a multiplicative constant ec (p 2 )=cb (p 2 ).5..5 To be fixed by a physical boundary condition...

13 Running gluon mass Return to the ST identity: q µ e np µ (q, r, p)+ e C (q, r, p) =i (r) i (p), (q 2 )=q 2 J(q 2 )+m 2 (q 2 ) inetic term running mass ec (q, r, p) =m 2 (r 2 )P (r) m 2 (p 2 )P (p) ec (q, r, p) =m 2 (r 2 ) m 2 (p 2 ) Expand around q = ec (r 2 )= m2 (r 2 ) dr 2 Integration m 2 (x) =m 2 () + c Z x dy B (y)

14 In order for m 2 (x) to admit a running gluon mass interpretation : m 2 (x) must be a monotonically decreasing function c = c fixes the sign m 2 (x) =m 2 () c Z x dy B (y) m 2 () = m 2 () = c Z dy B (y) power-law m 2 (x) M 4 x +p fixes the modulo.6.4 Finally: m 2 (x) = c Z x dy B (y).2...

15 The SDE-BSE system (with. D.Binosi) Y ( 2 ) m 2 () = C B s A 2 s C = Z dyb (y) B = 3C A 8 F () Z dyy 2 2 (y)b (y) A = 3C2 A 32 3 F () Z A s 2 + B s + C = dyy 2 2 (y)y (y)b (y) Quadratic equation for s

16 Self-consistency requirement : SDE s = B + p B 2 4AC 2A! = s BSE for a given MOM subtraction point µ ; In our case, µ =4.3 GeV s To enforce the equality we must model the three-gluon vertex (with 3 quantum gluons) Ansatz: µ = f(q, r, p) () µ q µ r form factor Tree level tensor p

17 Infrared features of the the three-gluon vertex massless ghost propagators log-divergences in the infrared ghost loop 3 2 zero crossing symmetric point: q 2 = p 2 = r 2 f(q 2 )=a apple+bln q2 + m 2 µ 2 + c ln q2 µ β= = β= = β protected unprotected f()!

18 A. Cucchieri, A. Maas, and T. Mendes, Phys.Rev.D74, 453 (26); Phys.Rev. D77, 945 (28). A. Athenodorou, D. Binosi, P. Boucaud, F. De Soto, J. Papavassiliou, J. Rodriguez- Quintero, and S. Zafeiropoulos, Phys. Lett. B76, 444 (26). A. G. Duarte, O. Oliveira, and P. J. Silva, Phys. Rev. D94, 7452 (26). A. C. Aguilar, D. Binosi, D. Ibanez, and J. Papavassiliou, Phys. Rev.D89, 858 (24). A. Blum, M. Q. Huber, M. Mitter, and L. von Smeal, Phys. Rev. D89, 673 (24). G. Eichmann, R. Williams, R. Alofer, and M. Vujinovic, Phys. Rev. D89, 5 (24). A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowsi, and N. Strodthoff, Phys. Rev. D94, 545 (26). R. Alofer, M. Q. Huber, and K. Schwenzer, Phys. Rev. D8, 5 (2); Eur. Phys. J. C62, 76 (29). R. Williams, C. S. Fischer, and W.Heupel, Phys. Rev. D 93, no. 3, 3426 (26).

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20 Additional material

21 Second ingredient: Seagull identity In regularization schemes that preserve translational invariance, such as dimensional regularization : Z 2 + d 2 Z f( 2 )= One-loop example: scalar QED Z 2 ( 2 + m 2 ) 2 = (4 ) d 2 Z 2 + m 2 =(4 ) d 2 µ (q) = d 2 µ d (m 2 ) d 2 2 q + q d (m 2 ) d 2 2 ν + µ ν Z 2 ( 2 + m 2 ) 2 + d 2 Z Seagull 2 + m 2 = identity f( 2 )= 2 + m 2