Effective potential at 3 loops Stephen P . Martin Northern Illinois University LoopFest XVIII Fermilab, August 14, 2019 Based on 1709.02397, and code written with Dave Robertson: 1907.02500 SMDR 1610.07720 3VIL 1
Motivations The effective potential 1 1 1 V eff ( φ j ) = V (0) + 16 π 2 V (1) + (16 π 2 ) 2 V (2) + (16 π 2 ) 3 V (3) + . . . is useful for: • Addressing (meta-)stability of the Standard Model electroweak vacuum • Relating the Standard Model VEV to the Lagrangian parameters m 2 , λ, y t , g 3 , g, g ′ , . . . ⋆ Typically, eliminate m 2 in favor of VEV with high precision. • Precise treatment of spontaneous symmetry breaking in your favorite New Physics model ⋆ What symmetries are broken? ⋆ What are the scales of VEVs? I will report on the computation of V eff through full 3-loop order in a general renormalizable theory, and specialization to the Standard Model. 2
In the Standard Model at tree level: V (0) = m 2 | Φ | 2 + λ | Φ | 4 Easy recipe: the rest of the effective potential is computed as the sum of 1-particle-irreducible vacuum (no external legs) Feynman diagrams in Landau gauge, with masses and couplings derived with a constant scalar background field. In electroweak perturbation theory, expand the Higgs field about the VEV: v √ Φ = + H 2 where v is a constant background field of order 246 GeV. But there are at least two distinct ways that this is commonly done. . . 3
Two common definitions of the VEV: � − m 2 /λ • Tree-level VEV: v tree = – Advantage: manifestly gauge-invariant – Disadvantage: must include tadpole graphs, perturbation theory includes factors 1 /λ n at loop order n • Loop-corrected VEV: v = minimum of the full effective potential – Advantage: tadpole graphs vanish, need not be included. Sum of all Higgs tadpoles ∝ ∂V eff /∂φ = 0 . – Disadvantage: depends on gauge choice; at 3-loop order, only tractable in Landau gauge. (See SPM and Hiren Patel, 1808.07615, for 2-loop order V eff with general gauge fixing.) The first definition is often used, but I prefer the “tadpole-free” scheme following from expanding the Higgs field around the VEV v in the second definition. 4
The problem with tadpoles: 1 1 h = p 2 + m 2 2 λv 2 Perturbation theory converges more slowly if one expands the Higgs field around the tree-level VEV. For observables (such as pole masses, G F , etc.), the leading loop-expansion parameter is N c y 4 t (expand around v tree , need tadpoles) 16 π 2 λ N c y 2 t (expand around v , tadpoles vanish) . 16 π 2 5
Coleman-Weinberg (1-loop) effective potential in MS scheme: � ( M 2 ) 2 � � V (1) ( φ ) ln( M 2 ) − 3 / 2 = 4 real scalars � ( M 2 ) 2 � � ln( M 2 ) − 3 / 2 − 2 4 Weyl fermions � ( M 2 ) 2 � � ln( M 2 ) − 5 / 6 + 3 4 real vectors where ln( x ) = ln( x/Q 2 ) , with Q = renormalization scale, and M 2 = MS squared mass, dependent on background field φ . Beyond 1 loop, φ also enters through field-dependent couplings. . . 6
Topologies of loop corrections to the effective potential: V (1) = Coleman and E. Weinberg V (2) = + Ford, Jack, Jones hep-ph/0111190 V (3) = 1 1 1 2 5 1 4 2 3 4 1 2 3 4 6 5 3 4 5 3 4 5 6 3 4 2 2 2 1 3 E 1234 G 12345 H 123456 J 12345 K 123456 L 1234 After taking into account symmetries and gauge invariance, the V (3) for a general renormalizable field theory can be written in terms of 89 loop integrals. 7
Examples: x u x v y u z v z y w w H F F SV FF ( u, v, w, x, y, z ) K V V SSF F ( x, w, u, z, y, v ) Propagator labels: • S = scalar • F = helicity-preserving fermion • F = helicity-violating fermion (mass insertion) • V = vector 8
H SSSSSS , K SSSSSS , J SSSSS , G SSSSS , L SSSS , E SSSS , H F F F SSS , H F F F SSS , H F F SSF F , H F F SSF F , H F F SSF F , H F F SSF F , K SSSSF F , K SSSSF F , K F F F SSF , K F F F SSF , K F F F SSF , K F F F SSF , K F F F SSF , K SSF F F F , K SSF F F F , K SSF F F F , J SSF F S , J SSF F S , H SSSSSV , H V V SSSS , H SSV V SS , H V V V SSS , H SSSV V V , H V V SSV S , H SSV V V V , H SV V V SV , K SSSSSV , K SSSSV V , K SSSV V S , K V V SSSS , K SSSV V V , K V V SSV S , K SSV V V V , K V V SV V S , J SSV SS , J SSV V S , G V SV V S , H gauge , S , K gauge , S , K gauge , SS , H F F V V F F , H F F V V F F , H F F V V F F , H F F V V F F , H F F F V V V , H F F F V V V , K F F F V V F , K F F F V V F , K F F F V V F , K F F F V V F , K F F F V V F , K V V F F F F , K V V F F F F , K V V F F F F , K gauge , F F , K gauge , F F , H F F SV F F , H F F SV F F , H F F SV F F , H F F SV F F , H F F SV F F , H F F F V SS , H F F F V SS , H F F F V SS , H F F F SV V , H F F F SV V , H F F F SV V , K F F F SV F , K F F F SV F , K F F F SV F , K F F F SV F , K F F F SV F , K F F F SV F , K SSSV F F , K SSSV F F , K SSV V F F , K SSV V F F , K V V SSF F , K V V SSF F , K V V SV F F , K V V SV F F , H gauge , K gauge . These 89 functions each depend on squared mass arguments x, y, z, . . . and the MS renormalization scale Q . Most are far too lengthy to be given in print, so are provided in an electronic file in terms of 3-loop basis vacuum integrals, which have to be computed numerically. 9
Need to be able to systematically compute hundreds of integrals, for example: Standard Model Supersymmetry ˜ ˜ ˜ t t N j W W W t i C i g, ˜ ˜ h b N i W,H ± ˜ ˜ ˜ Z, h Z, h t W b j N n b C k W, H ± W,H ± Z, h, γ W In Standard Model case, the mass hierarchies are not all large. In SUSY cases, the mass hierarchies not known in advance. Reduce to basis (“master”) integrals, compute numerically using differential equations in the squared mass arguments. 10
The basis integrals for 3-loop vacuum diagrams are: x y x Known analytically, present no problems. z A ( x ) I ( x, y, z ) and genuinely three-loop scalar integrals: u v u v x z y y y u z v z w w H ( u, v, w, x, y, z ) G ( w, u, z, v, y ) F ( u, v, y, z ) which are known in 1-scale and some 2-scale special cases, but in general require numerical computation. Any 3-loop vacuum integral can be written as linear combinations of these, with coefficients that are rational functions of the squared masses, obtained using integration-by-parts identities. 11
The generic case: consider the master tetrahedral topology, and all corresponding basis integrals obtained by removing propagator lines: H ( u, v, w, x, y, z ) , G ( w, u, z, v, y ) , G ( x, u, v, y, z ) , G ( u, v, x, w, z ) , G ( y, v, w, x, z ) , G ( v, u, x, w, y ) , G ( z, u, w, x, y ) , F ( w, u, x, y ) , F ( w, v, x, z ) , F ( x, u, w, y ) , F ( x, v, w, z ) , F ( u, v, y, z ) , F ( u, w, x, y ) , F ( y, u, v, z ) , F ( y, u, w, x ) , F ( v, u, y, z ) , F ( v, w, x, z ) , F ( z, u, v, y ) , F ( z, v, w, x ) , products of I and A functions The derivatives of all of these with respect to any squared mass argument u, v, w, x, y, z are also 3-loop integrals, and so are linear combinations of the basis. Solve differential equations in the masses to compute these using Runge-Kutta, starting from known analytical values at a fixed reference squared mass a as initial conditions: H ( a, a, a, a, a, a ) , G ( a, a, a, a, a ) , F ( a, a, a, a ) , I ( a, a, a ) , A ( a ) . 12
3VIL = 3-loop Vacuum Integral Library SPM and Dave Robertson, 1610.07720 • Written in C, can be called from C, C++, Fortran, . . . • Uses analytic results where available, otherwise differential equations method • Evaluation for generic mass inputs: – Time < 1 second for generic cases on reasonably modern hardware – Relative accuracy < ∼ 10 − 10 – When computing a basis integral H ( u, v, w, x, y, z ) , simultaneously computes all subordinate basis integrals formed by removing propagators. • See also TVID (S. Bauberger and A. Freitas), uses dispersion relations. 13
In the Standard Model, the field-dependent squared masses that enter into the computation of V eff are: Z = ( g 2 + g ′ 2 ) v 2 / 4 , y 2 t v 2 / 2 , W = g 2 v 2 / 4 , t = m 2 + 3 λv 2 , G = m 2 + λv 2 . H = A problem: the Goldstone boson squared mass G can be very small, or negative. 1) If G < 0 , then V eff is complex even at 1-loop, due to terms with ln( G ) . Usually, a complex V eff means instability, but there is no physical instability here. 2) If G → 0 , then starting at 3-loop order, get infrared divergences in V eff . Need to make sure IR divergences do not infect physical observables. 14
Goldstone boson tree-level (mass) 2 G as a function of renormalization scale Q , at minimum of V eff : For Q > ∼ 100 GeV, we really do have tachyonic Goldstones: G < 0 . 15
The Goldstone Boson Catastrophe The leading behavior as G → 0 is: V (1) ∼ 3 4 G 2 ln G, 2nd derivative singular as G → 0 � � V (2) ∼ − 3 N c y 2 ln t − 1 1st derivative singular as G → 0 t t G ln G, � � 2 ln G. V (3) ∼ 3 N c y 2 t t (ln t − 1) singular as G → 0 with t, G = squared masses of top, Goldstone. These come from diagrams: G 0 G ± G 0 G ± t t t t t t b b t b G 0 ,G ± G ± G 0 t t At higher loop orders, the G → 0 singularities get worse. . . 16
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