One-Loop Integrals at vanishing external momenta and appli. Higgs potentials reconstructions CALC Mikhail Dolgopolov. Samara State University

Transcription

1 One-Loop Integrals at vanishing external momenta and applications for extended Higgs potentials reconstructions CALC 2012 Samara State University

2 1. Basic examples and an idea for extended Higgs potentials analysis by catastrophe theory methods [Dolgopolov, Dubinin, Rykova, Journal of Modern Physics, 2011] 2. Basic examples for one-loop integrals need for nite-temperature and zero-temperature eective Higgs potential reconstruction 3. Higgs potential bifurcation sets in MSSM and NMSSM (Elena Petrova talk)

3 Effective potential General form U eff Μ Μ Μ Μ Λ 1 T Λ 2 T Λ 3 T Λ 4 T Λ 2 5 T Λ 2 5 T Λ 6 T Λ 6 T Λ 7 T Λ 7 T Vacume averages Ν Ν 2 Effective potential in terms of vevs: U eff 1 Μ Ν Μ Ν 2 2 ReΜ 2 12 Ν 1 Ν 2 Λ 1 Λ 2 Ν Λ Ν 1 2 Ν 2 2 ReΛ 6 2 where Λ 345 Λ 3 Λ 4 ReΛ 5 Ν 1 3 Ν 2 ReΛ 7 2 Ν Ν 3 2 Ν 1,

4 Example

5 Example

6 Minimization conditions Equivilibrium Stationary conditions U eff 0 Μ 2 1 ReΜ2 Ν 2 12 Λ Ν 1 Ν Λ Ν ReΛ 2 6 Ν 1 Ν 2 1 ReΛ 2 7 Ν 2 Μ 2 2 ReΜ2 Ν 1 12 Λ Ν 2 Ν Λ Ν ReΛ 2 7 Ν 1 Ν 2 1 ReΛ 2 6 Ν 1 Type of equilibrium is defined by Hessin 3 Ν 1 3 Ν 2

7 Classic analogy

8

9 Morse lemma Two-doublet potential evolution depends on two fields Ν 1 & Ν 2 evolution and 7 parameters Λ i i Local propities of potential could be described by methods of catastroph theory. We use Morse lemma for effective potential to rewrite one in canonical form. U eff 0 det 2 U eff Ν i Ν j 0 U eff U canon Μ 1 Ν 1 2 Μ 2 Ν 2 2 Μ 1,Μ 2 Hessian eigenvalues Ν 1,Ν 2 new variables U eff and U canon are quality equivalent

10 in different variables Nonlinear transformations Transform potential due to Morse lemma: Linear transformation, diagonalization Ν 1 Cos Θ v 1 Sin Θ v 2 Ν 2 Sin Θ v 1 Cos Θ v 2 1 Μ v Μ v 2 1 ReΜ 2 12 v 1 v 2.. Μ 2 1 v Μ v where Μ 1,2 = 1 2 Μ 2 1 Μ 2 2 ± Μ 2 1 Μ Μ4 12 Nonlinear transformation (axis-conserve) V1 v 1 A 20 v2 1 A11 v 1 v 2 A 02 v2 2 A 30 v3 1 A21 v 2 1 v 2 A 12 v 1 v2 2 A03 v3 1 V 2 v 2 B 20 v2 1 B11 v 1 v 2 B 02 v2 2 B 30 v3 1 B21 v 2 1 v 2 B 12 v 1 v2 2 B03 v3 1

11 Equal coefficients for same powers of variables. In system of nonphysical fields potential is presented in unique form, which defines stable equivilibrium state. U eff v 1, v 2 U canon V1, V2 Μ 1 v 1 2 Μ 2 v Μ 1 V1 2 Μ 2 V2 2

12

13 R.P.Feynman Phys.Rev.76: ,1949. A. J. Buras, P. H. Chankowski, J. Rosiek and L. Slawianowska, Nucl. Phys. B 659, 3 (2003) [arxiv:hep-ph/ ]. Athanasios Dedes, Janusz Rosiek, and Philip Tanedo, PHYSICAL REVIEW D 79, (2009). L. Vergara, Evaluating one loop integrals at nite temperature. Journal of Physics A30 (1997) P.Amore, Non perturbative regularization of one loop integrals at nite temperature. Journal of Physics A38 (2005) (hep-th/ ) P. Amore, Convergence acceleration of series through a variational Journal of Mathematical Analysis and

14 Applications. Journal of Mathematical Analysis and Applications 323 (2006) 6377 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, NY, 1964 E. Akhmetzyanova, M. Dolgopolov, M. Dubinin, Higgs bosons in the two-doublet model with CP violation. Phys.Rev. D71 (2005) (hep-ph/ ) E. Akhmetzyanova, M. Dolgopolov, M. Dubinin, Violation of CP Invariance in the Two-Doublet Higgs Sector of the MSSM. Phys.Part.Nucl. 37 (2006) M. Laine, Eective theories of MSSM at high temperature. Nucl.Phys. B481 (1996) 4384 (hep-ph/ ); Erratum-ibid. B548: , 1999

15 G.R. Farrar, M. Losada, SUSY and the electroweak phase transition. Phys.Lett. B406 (1997) 6065 (hep-ph/ ) M. Losada, High temperature dimensional reduction of the MSSM and other multiscalar models. Phys.Rev. D56 (1997) (hep-ph/ )

16 The smallness of external state masses relative to the masses of loop particles allows us to take the limit where the external particles are massless. These integrals are particularly easy to solve using algebraic recursion relations or residue method Remark. But it is necessary to keep the external momentum p μ general (i. e. nonzero) until it can be converted into an external mass in the amplitude, after which point one may take accurate limit p 0, especially in the cases with equivalent internal masses in loop.

17 In d dimensions such two-point loop integral B 0 is dened as: 1 (4π) B 0(m 2, M 2 ) = μ 4 d 2 d d k i (2π) d (k 2 m 2 ) [k 2 M 2 ]. (1) In order to derive asymptotic results we expand B 0 (p 2, m 2 1, m2 2 ) in powers of the external momentum p 2 : B 0 (p 2, m 2 1, m 2 2) = μ m2 1 log m2 1 m2 2 log m2 2 m 2 2 m2 1 ( +O p 2 m m2 2 ).

18 The 3- and 4-point loop integrals at vanishing external momenta are dened as: 1 (4π) C 2n(m1, 2 m2, 2 m3) 2 = μ 4 d 2 1 (4π) 2 D 2n(m 2 1, m 2 2, m 2 3, m 2 4) = μ 4 d d d k (2π) d ik 2n 3 i (k 2 m 2 i ),(2) d d k ik 2n (2π) d 4 i (k 2 mi 2).(3) Note that the above equations are manifestly equivalent under interchange of the arguments and explicit calculations show that one may reshue the order of the arguments without changing the result (for functions with unequal arguments it is obvious, and for functions with the equivalent arguments it demands accuracy!).

19 The two-point loop integral in tensor reduction B 1 can be dened as: μ 4 d 1 (4π) 2 p μb 1 (p, m 2, M 2 ) = (4) d d k ik μ (2π) d (k 2 m 2 ) [(k + p) 2 M 2 ]. (5) Such integrals appear in the self-energy contributions.

20 It turns out that this is very simple to evaluate integrals (1), (2), (3) with the following algebraic relations: ( 1 (k 2 m 2)(k 1 2 m 2) = 1 2 m 2 1 m k 2 m 2 1 k 2 m 2 2 ), (6) k 2 k 2 m = 1 + m2 2 k 2 m. (7) 2 Note that denominators of integrals under consideration are spherically symmetric.

21 And for the case of all equivalent masses the particularly well-known general integral can be performed using Γ function: d n k (k 2 ) b (2π) n (k 2 A 2 ) a = (8) i (4π) n/2 ( 1)a+b (A 2 b a+n/2 Γ(b + n/2)γ( b + a n/2) ), (9) Γ(n/2)Γ(a) where the left-hand side is an integral in n-dimensional Minkowski space. In the private case at b = 0 d n k 1 (2π) n (k 2 A 2 ) a = i (4π) n/2 ( 1)a (A 2 a+n/2 Γ(a n/2) ). Γ(a) (10)

22 The explicit formulae at d 4 are listed below (we give also expressions for some 3-point functions proportional to higher momenta powers, may be useful in contributions for Higgs potential calculations) and may be represented only in terms of B 0 integral, or A 0 (10):

23 1 (4π) A 0(m 2 ) = μ 4 d 2 d d k i (2π) d k 2 m = (11) 2 μ 4 d i = i (4π) d/2 ( 1)(m2 ) 1+d/2 Γ(1 d/2) =(12) = i i ( 1 + ε ) 16π 2 2 log(4πμ2 ) + O(ε 2 ) m 2 (13) ( 1 ε ) 2 log(m2 ) + O(ε 2 ) ( ) 2 ( 1) ε + 1 γ E + O(ε) A 0 (x) = x( μ + log x), (14)

24 B 0 (x, y) = 1 x y [A 0(x) A 0 (y)] = (15) = 1 x y [x μ y μ x log x + y log y] = = x log x + y log x y log x + y log y μ = x y = + log x μ y 2 x y log y x = (16) = B 0 (y, x) = + log y μ x 2 y x log x y, (17)

25 1 C 0 (x, y, z) = y z [B 0(x, y) B 0 (x, z)] (18) y log y z z log = x (x y)(z y) + x (x z)(y z), (19) C 0 (x, y, y) = y B 0(x, y) = 1 x log x y 1 +, (20) y x y x C 2 (x, y, z) = B 0 (x, y) + zc 0 (x, y, z) = y z y 2 log z 2 log = log μ2 x + x (x y)(z y) + x (x z)(y z),

26 C 2 (x, y, y) = B 0 (y, x) + yc 0 (x, y, y) = = μ + log y x y x log x y + = log μ2 y x x y y y x x y x log x y x y 1 + x log x y y x = (21)

27 B 0 (x, y) + yc 0 (x, y, y) = = log μ2 x + y y x 1 = y A 0(y) + xc 0 (x, y, y) = = log μ2 y x x y (y 2x) log x y y x x y x log x y x y, (22). (23) The answers (21), (23), and (22) are identical indeed, check by Mathematica is done. Check for last equations:

28 1 (4π) 2 m A 0(m 2 ) = μ 4 d 2 d d k (2π) d i (k 2 m 2 ) 2 = (24) μ 4 d i = i (4π) d/2 ( 1)2 (m 2 ) 2+d/2 Γ(2 d/2) = i i ( = 1 + ε ) 16π 2 2 log(4πμ2 ) + O(ε 2 ) (25) ( 1 ε ) 2 log(m2 ) + O(ε 2 ) ( ) 2 ε γ E + O(ε) = ( μ + 1) + log x yes x A 0(x) = μ + log x + 1. (26)

29 Equal arguments correspond to equivalent masses. The divergent part is {μ; 1} = 2 4 d + log(4π{μ2 ; 1}) γ E + 1 where μ is the renormalization scale. And C 11 (x, y) = x 3y 4(x y) 2 + y 2 2(x y) 3 log y x, (27) C 12 (x, y) = x + y 2(x y) 2 xy 3(x y) 3 log y x. (28)

30 The explicit formulae for the four-point integrals at vanishing external momenta are (possible cross-checks are provided): D 0 (x, y, z, t) = 1 z t (C 0(x, y, z) C 0 (x, y, t)) = = + y log y z log z x + x (y x)(y z)(y t) (z x)(z y)(z t) + (29) t t log x (t x)(t y)(t z), (30)

31 D 0 (x, y, z, z) = z C 0(z, y, x) (31) = y log y x x log z z (z y)(x y) + z = (z x)(y x) = x y 1 x log y log (x z)(y z) + z (x y)(x z) + z 2 (y x)(y z), 2

32 And some additional private checks: B 0 (m 2 ) = m 2 A 0(m 2 ) = μ + log m 2 + 1, (32) C 0 (m 2 ) = 1 2 m B 0(m 2 ) = 1 2 2m, 2 (33) D 0 (m 2 ) = 1 3 m C 0(m 2 ) = 1 2 6m. 4 (34)

33 [CALC'2009] In the nite temperature eld theory Feynman diagrams with boson propagators, containing Matsubara frequencies ω n = 2πnT (n = 0, ±1, ±2,...), lead to structures of the form I [m 1, m 2,..., m b ] = T n= dk (2π) 3 b ( 1) b (35) (k 2 + ωn 2 + mj 2 ), Here k is the three-dimensional momentum in a system with the temperature T. In the following calculations rst we perform integration with respect to k and then take the sum, using the reduction to three-dimensional theory in the high-temperature limit for zero frequencies. I [m 1, m 2,..., m b ] = 2T (2πT ) 3 2b ( 1) b π 3/2 Γ(b 3/2) S(M, b 3/2), (2π) 3 Γ(b) (36) i=1

34 where S(M, b 3/2) = {dx} n=1 1 ( m ) (n 2 + M 2 ), M 2 2. b 3/2 2πT (37) For b > 1 the parameter m 2 is a linear function dependent on m 2 i and the variables {dx} of Feynman parametrization, which are the integration variables in (37). At the integer values of b the integrand in (3) is a generalized Hurwitz zeta-function Note that for the leading threshold corrections to eective parameters of the two-doublet potential b > 2, so the wave-function renormalization appears in connection with the divergence at b = 2

35 M 2 (M a, M b, x) = (Ma 2 Mb 2)x + M b 2. Then we get 1 [k 2 + ma][k mb 2] = 1 (2πT ) dx (p 2 + n 2 + M 2 ) 2. (54) and divergent series for (45) (dk = (2πT ) 3 dp) I 0 [M a, M b ] = 1 1 dp 1 dx 2πT (2π) 3 (p 2 + n 2 + M 2 ), 2 0 n=,n 0 (55) With the help of dimensional regularization or dierentiating the integral dp 1 (2π) 3 (p 2 + M 2 ) = M 2 4π + O(M T ) (56) 2 over the parameter M, the equation (55) can be reduced to 1 I 0 [M a, M b ] = 16π 2 T 1 dx ζ(2, 1 2, M 2 ), (57) 0

36 where ζ(u, s, t) is the generalized Hurwitz zeta-function 1 ζ(u, s, t) = n=1 1 (n u + t) s. (58) So in the case under consideration the sums of integrals (51) and (52) can be calculated by dierentiation of (57) with respect to mass parameters participating in M = M(M a, M b, x). Dierentiation increases the power s in the denominator of (57) giving convergent integrals I 1 [M a, M b ] = T I 0 = 1 2M a M a 64π 4 T 2 I 2 [M a, M b ] = 1 ( I 1 ) = 2M b M b π 6 T 4 dx x ζ[2, 3 2, M 2 (x)], 1 0 (59) dx x (1 x) ζ[2, 5 2, M 2 (x) (60)

37 3. Temperature evolution. Summary 1. Method for stable state research 2. Constrains on parameters.