The unitary pion mass at Op 4 ) in SU) ChPT Chris Kelly ebruary, This section contains the derivation of the behaviour of the pion mass, specifically the mass of the φ 3 = π, at NLO in SU) ChPT. The partially-quenched version of this form is used to perform chiral extrapolations of lattice data in sections.... At Op 4 ) Weinberg s power counting equation D = 4) gives 4 = +N L + N 4, ) where N L is the number of loops and N 4 is the number of vertices from L 4. Here the terms in N n = N 6... have been set to zero as the coefficients n for n > cannot satisfy the equation for positive integer N n. The coefficient of N is always zero. Equation shows that at Op 4 ) the pion self-energy receives corrections at -loop from L N L =,N 4 = ) and tree-level corrections from L 4 N L =,N 4 = ). In the first subsection the L -loop corrections are determined: these give rise to the chiral logarithms. The second subsection contains the tree-level L 4 corrections which the last subsection collects along with the -loop corrections to derive the mass behaviour.. -loop corrections from L The leading order Lagrangian L has the form L = 4 Tr µ U µ U ) + 4 Tr χu +Uχ ), ) where Ux) = exp i φx) ) φ = φ 3 φ iφ φ + iφ φ 3 = π π + π π, 3)
and χ = BmI. The diagrams of interest have the form iσ L p ) =, 4) L where p is the momentum of the incoming and outgoing φ 3 particles and k is the loop momentum. The loop can contain any of the three components of φ. Expanding L in U up to fourth order U = I+ i φ ) + i φ ) + i φ ) 3 + i φ ) 4 +Oφ 5 ) 5)! 3! 4! and keeping only the terms containing two φ 3 fields, one obtains the standard kinetic and mass terms L free = 3 µ φ 3 Bmφ 3 6) and a set of interaction terms which will be discussed shortly. rom eqn. 6 the leading order term in the pion mass can be seen to be M LO φ 3 ) = Bm. 7) The same terms arise for the other components of φ, thus the Goldstone bosons are degenerate. If the quark mass degeneracy is broken through modifying the mass matrix χ Bdiagm u,m d ), 8) the mass becomes M LO,ndeg φ 3 ) = Bmu + m d ). 9) Again the same terms arise for the other components of φ, thus the LO φ mass degeneracy is not lost if the quarks are not degenerate. or simplicity and in order to have use in current lattice simulations, the remainder of this calculation is performed with degenerate quarks. The interaction Lagrangian of L contains no three-point terms or terms in any odd power of φ). This is because L is symmetric under a global sign-flipping φ i
φ i i: under this transformation U = exp i φ ) exp i φ ) = U, ) and U U due to the Hermiticity of φ, hence the kinetic term remains invariant. Due to the real-diagonal nature of the mass matrix it is also Hermitian, thus the terms in χ are also invariant. After expanding in φ, any term with an odd power of φ picks up a minus sign under the sign-flip and are thus forbidden due to the global invariance. Note that the NLO Lagrangian L 4 does not possess this symmetry and thus interactions with off powers of φ are allowed. The four-point terms contain either two- or zero- factors of the momentum. Those with momentum dependence arise from the µ U) term, and contain only two factors of the momentum because each term in µ U = i ) +! ) φ i ) + 3 3! ) φ i ) + 4 4! ) φ 3 i ) +... ) contains only a single factor of µ φ. The other terms arise from the remainder of L which has no derivative terms and thus the terms have no momentum dependence. Here the four-point terms are separated according to their field content, and momentumspace eynman rules are derived following the usual procedure of multiplying by i, replacing partial derivatives µ φ i p i ) with ip i and symmetrising over permutations of like fields. All momenta are considered to be flowing towards the vertex. Bmφ 4 3 = 4!i i Bm = Bm 6 µ φ 3 ) φ j φ 3 j ) + µ φ 3 )φ 3 µ φ j )φ j + Bmφ 3 φ j ) L = i 3 p q+r s ) p+q) µr+s) µ + Bm. Here the index j {, }. The symmetry factor of the first diagram arises from the 4! L ) 3
ways of associating a field φ 3 with a leg of the vertex. Similarly, symmetry factors of 4 arise from the ways of arranging the fields in the first, second and last term of the second diagram. Using the eynman rules above, with q = p and r = s = k momenta are considered incoming in the eynman rules) and using the fact that the free-field propagators of the φ fields are identical, the three diagrams of eqn. 4 can be summed as iσ L p ) = = 4 3 d 4 k i π) 4 k Bm i d 4 k π) 4 k Bm i Bm+ p 3 k + Bm ) ) p k + 5 ) 3) Bm Dimensionally regularising to d-dimensions, and after a Wick rotation p ip, p p to Euclidean space, this becomes iσ L p ) = 4i d d k p + 5 Bm) 3 π) d k + Bm + ) d d k k π) d k. 4) + Bm Transforming to 4-spherical coordinates, the integral of the first term of the above becomes d d k π) d k + M = dω d dl π) d l d l + M, 5) where M = Bm. Defining x = M l + M eqn. 5 becomes such that l = M x ) and dl = + M l ) lm dx, π) d M ) d which contains the the Beta function dω d dxx d x) d 6) Bα,β) dxx α x) β = Γα)Γβ) Γα + β) 7) with α = d and β = d. The angular integral is a standard result dω d = π) d Γ d. 8) 4
Equation 6 therefore integrates to π) d π) d M ) d Γ d ) Γ) = 4π) d M ) d Γ d ), 9) using Γ) =! =. With d = 4 ε for small ε, the expansion of Γ + ε ) about its pole at has the form Γ + ε ) = + γ +Oε). ) ε Using the result X ε = +ε logx)+oε ) and introducing a scale factor µ ε to maintain the canonical dimensionality of the couplings, eqn. becomes M 4π) µε 4π) ε M ) ε ε Γ + ) = M ) 4π) +ε log µ +Oε ) ε ) log4π +Oε ) + ε ) logm +Oε ) ε ) + γ +Oε) = M 4π) ) ε + γ log4π + log M µ +Oε). }{{} IM, µ,ε) ) Transforming back to Minkowski space p p, the first term of eqn. 4 thus becomes Treating the second term of eqn. 4 as above 4iM p + 5 ) 4 M 34π) IM, µ,ε). ) d d k k π) d k + M = π) d Γ d ) = M ) d 4π) d Γ d ) dl l d+ π) d l + M dxx d x) d = M ) d 4π) d Γ d ) Γ d )Γ+ d ) }{{} d Γd ) = M ) d 4π) d d Γ d ). 3) 5
Again setting d = 4 ε, introducing a scale dependence µ ε and using Γ +ε) = ε γ + 3 +Oε), 4) 4 eqn. 3 becomes M ) 4π) IM,µ,ε). 5) Combining this with eqn., eqn. 4 is thus iσ L p ) = im 64π) IM, µ,ε) M 4p ). 6) Here an overall symmetry factor of which been applied. This arises from the fact that for any choice of two legs A and B to contract together to form the loop, there are two identical assignments: A B and B A which have until now been overcounted.. Tree-level corrections from L 4 The NLO Lagrangian L 4 without has the form L 4 = l { Tr [ µ U µ U) ]} + l Tr [ µ U ν U) ] Tr [ µ U ν U) ] +l 3 Tr [ µ U µ U) ν U ν U) ] + l 4 Tr [ µ U µ U) ] Tr χu +Uχ ) +l 5 Tr [ µ U µ U) χu +Uχ )] + l 6 [ Tr χu +Uχ )] +l 7 [ Tr χu Uχ )] + l8 Tr χu χu +Uχ Uχ ), where l i are low-energy constants. Expanding U up to second order in φ, keeping only the two-point interactions of φ 3 and applying the rules of the previous section, one finds a two-point vertex with the eynman rule ) p φ 3 φ = 6iBm [ p 3 L 4 l 5 + l 4 ) 4Bml 6 + l 8 ) ] = iσ L4 p ), 8) which forms the L 4 correction to the self-energy. 7) 6
.3 Calculation of the pion mass The corrections to the self-energy can be considered as an additive renormalisation of the pole mass in the dressed propagator. The dressed propagator has the form k M ΣM ), 9) where ΣM ) is the sum over the -PI connected diagrams. In the previous two subsections, Σ has been calculated to order p 4. These solutions have the form Σp ) = A+Bp 3) where and A = M ) 64π) IM, µ,ε)+ 6M ) l 6 + l 8 ) 3) B = M 34π) IM, µ,ε) 8M l 5 + l 4 ). 3) Thus with M Op 4 ) = M +B)+A = M + 8M l 8 l 5 )+ 6M l 6 l 4 ) + M ) 4π) IM, µ,ε), IM, µ,ε) = ) ε + γ log4π + log M µ +Oε) 33) 34) as before. If renormalised LECs are defined in an MS-fashion l r i = l i Γ i ε ) 4π) + γ log4π 35) where Γ i are rational coefficients, the NLO pion mass obtains its standard form M Op 4 ) = M + 8M l 8 l 5 )+ 6M M l 6 l 4 )+ 4π) log M ) µ 36) Note that many texts use a modified decay constant f = rather than the used here. The coefficients Γ i for i = {4,5,6,8} can be calculated by considering three other processes at Op 4 ) in the chiral expansion, but that is beyond the scope of this work. 7